ACADEMIC    ALGEBRA 


B¥ 


WILLIAM   J.    MILNE,  Ph.D.,  LL.D. 

PRESIDENT  OF  NEW  YOEK  STATE  NORMAL  COLLEGE,  ALBANY.  N.Y. 


NEW   YORK . :  •  CINCINNATI  • : .  CHICAGO 

AMERICAN    BOOK    COMPANY 


1^  ^..  . 


COPTBIGHT,    1901,    BY 

WILLIAM  J.  MILNE. 
Entered  at  Stationers'  Hall,  London. 


ACADEMIC   algebra. 
E-P9 


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PURPOSE   AND   PLAN   OF   THIS   BOOK 

The  Academic  Algebra  has  been  prepared  to  meet  the  require- 
ments of  the  most  exacting  entrance  examination  of  any  College 
or  University  in  the  United  States. 

The  book  contains  a  thorough  treatment  of  the  science,  so  far 
as  it  is  taught  in  the  secondary  schools.  A  full  development  of 
each  subject,  and  a  clear  statement  of  its  principles  and  laws,  pre- 
cedes the  proofs  of  the  principles  —  an  arrangement  that  makes 
it  possible  for  a  teacher,  without  hindrance  to  the  progress  of  the 
student,  to  postpone,  if  he  sees  fit,  the  rigorous  proofs.  The 
examples  and  problems  are  sufficiently  numerous  and  complex 
to  test  the  student's  skill  in  applying  all  the  principles  that  are 
developed.  They  are  carefully  graded,  increasing  in  difficulty  in 
each  subject,  so  that,  if  desired,  a  brief  and  easier  course  may 
be  conveniently  provided  by  omitting  the  more  difficult  problems 
at  the  end  of  each  list. 

In  several  respects,  the  order  of  the  topics  deviates  from  that 
which  is  usually  followed.  These  innovations,  made  in  accord- 
ance with  sound  pedagogical  principles,  will  arouse  and  sustain  a 
greater  interest  in  the  science.  The  method  of  presentation  also 
is  unique.  The  principles  are  developed  by  appropriate  questions 
designed  to  lead  the  student  to  infer  and  apprehend  clearly  the 
truth  that  is  presented ;  these  are  followed,  first,  by  a  brief,  yet 
clear  and  complete  statement  of  the  principles,  and  then  by  full 
and  rigorous  proofs  of  the  principles.  Thus  the  natural  method 
of  mathematical  teaching  has  been  followed,  the  student  being 
led,  first,  to  make  proper  inferences ;  second,  to  express  the  infer- 
ences briefly  and  accurately ;  and,  third,  to  prove  their  truth  by 
the  method  of  deductive  reasoning. 

The  acknowledgments  of  the  author  are  due  to  Prof.  J.  H. 
Tanner,  of  the  Department  of  Mathematics,  Cornell  University, 
for  many  valuable  suggestions  in  connection  with  the  preparation 

of  this  book. 

WILLIAM  J.  MILNE. 
State  Normal  College,  Albany,  N.Y. 

3 

M1J26SJ4 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/academicalgebraOOmilnrich 


CONTENTS 


Algebraic  Solutions 7 

Definitions  and  Notation 12 

Addition 21 

Subtraction 32 

Multiplication 45 

Division 71 

Review 87 

Factoring         .        .        .        .       ^ 90 

Review  of  Factoring 108 

Highest  Common  Divisor 115 

Lowest  Common  Multiple 126 

Fractions 133 

Review  of  Fractions 160 

Simple  Equations ^    .  163 

Simultaneous  Simple  Equations 186 

Involution 214 

Evolution         ...        •        •        •        •        •        •        •        •        •  221 

Theory  of  Exponents 239 

Radicals 250 

Review 274 

Quadratic  Equations .  278 

General  Review 316 

Ratio  and  Proportion 325 

Variation 338 

5 


6  CONTENTS 

PAOlt 

Progressions c        ,        .        .        .        .  344 

Imaginary  and  Complex  Numbers 363 

Inequalities 372 

Variables  and  Limits 378 

Interpretation  of  Results 3&7 

Indeterminate  Equations 394 

The  Binomial  Theorem 398 

Logarithms .  405 

Undetermined  Coefficients          .......  422 

Permutations  and  Combinations 435 

Determinants 445 


ACADEMIC    ALGEBRA 


>J«ic 


ALGEBRAIC   SOLUTIONS 


1.  Problem  1.  A  man  had  400  acres  of  corn  and  oats.  If 
there  were  3  times  as  many  acres  of  corn  as  of  oats,  how  many 
acres  were  there  of  each  ? 

Arithmetical  Solution 
A  certain  number  =  the  number  of  acres  of  oats. 
Then,  3  times  that  number  =  the  number  of  acres  of  corn, 

and  4  times  that  number  =  the  number  of  acres  of  both  ; 

therefore,         4  times  that  number  =  400. 

Hence,  the  number  =  100,  the  number  of  acres  of  oats, 

and  3  times  the  number  =  300,  the  number  of  acres  of  com. 

Algebraic  Solution 

Let  X  =  the  number  of  acres  of  oats. 

Then,  3  a;  =  the  number  of  acres  of  corn, 

and  4  X  =  the  number  of  acres  of  both  ; 

therefore,  4x  =  400. 

Hence,  x  =  100,  the  number  of  acres  of  oats, 

and  3  X  =  300,  the  number  of  acres  of  corn. 

2.  An  expression  of  equality  between  two  numbers  or  quan- 
tities is  called  an  Equation. 

5  X  =  30  is  an  equation. 

3.  A  question  that  can  be  answered  only  after  a  course  of 
reasoning  is  called  a  Problem. 

4.  The  process  of  finding  the  result  sought  is  called  the  Solu- 
tion of  the  problem. 


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;8|/e%ff ';    r',]      :  jA^ADEMIC  ALGEBRA 

5.  The  expression  in  algebraic  language  of  the  conditions  of  a 
problem  is  called  the  Statement  of  the  problem. 

Solve  algebraically  the  following  problems: 

2.  A  horse  and  saddle  cost  $50.  If  the  horse  cost  4  times 
as  much  as  the  saddle,  what  was  the  cost  of  each  ? 

3.  A  bicycle  and  suit  cost  $90.  How  much  did  each  cost, 
if  the  bicycle  cost  5  times  as  much  as  the  suit  ? 

4.  Of  240  stamps  that  Harry  and  his  sister  collected,  Harry 
collected  3  times  as  many  as  his  sister.  How  many  did  each 
collect  ? 

5.  If  Mr.  Brown  and  his  son  together  had  $220,  and  Mr. 
Brown  had  10  times  as  much  as  his  son,  how  much  money  had 
each  ? 

6.  In  a  room  containing  45  students  there  were  twice  as  many 
girls  as  boys.     How  many  were  there  of  each  ? 

7.  A  had  7  times  as  many  sheep  as  B,  and  both  together  had 
608.     How  many  sheep  had  each  ? 

8.  A  and  B  began  business  with  a  capital  of  $  7500.  If  A 
furnished  half  as  much  capital  as  B,  how  much  did  each  furnish  ? 

Suggestion.  — Let  x  =  the  number  of  dollars  A  furnished. 

9.  A  man  bought  a  cow  and  a  calf  for  $  36,  paying  8  times  as 
much  for  the  cow  as  for  the  calf.     What  was  the  cost  of  each  ? 

10.  James  sold  his  pony  and  a  saddle  for  $  60.  If  the  saddle 
sold  for  I  as  much'  as  the  pony,  what  was  the  selling  price  of 
each? 

11.  A  certain  number  added  to  twice  itself  equals  96.  What 
is  the  number  ? 

12.  A  farmer  raised  a  certain  number  of  bushels  of  wheat,  4 
times  as  much  corn,  and  3  times  as  much  barley.  If  there  were 
in  all  4000  bushels  of  grain,  how  many  bushels  of  each  kind  did 
he  raise  ? 

13.  A  boy  bought  a  bat,  a  ball,  and  a  glove  for  $  2.25.  If  the 
bat  cost  twice  as  much  as  the  ball,  and  the  glove  cost  3  times  as 
much  as  the  bat,  what  was  the  cost  of  each  ? 


ALGEBRAIC   SOLUTIONS  9 

14.  In  a  fire  B  lost  twice  as  much  as  A,  and  C  lost  3  times 
as  much,  as  A.  If  their  combined  loss  was  §  300,  what  was  the 
loss  of  each  ? 

15.  A  house  and  lot  cost  $3000,  If  the  house  cost  4  times  as 
much  as  the  lot,  what  was  the  cost  of  each  ? 

16.  In  a  business  enterprise  the  joint  capital  of  A,  B,  and  C 
was  $  2100.  If  A's  capital  was  twice  B's,  and  B's  was  twice  C's, 
what  was  the  capital  of  each  ? 

17.  John,  William,  and  George  together  had  120  marbles.  If 
William  had  twice  as  many  as  John,  and  George  had  3  times  as 
many  as  John,  how  many  had  each  ? 

18.  In  an  orchard  of  apple,  pear,  and  cherry  trees,  containing 
1690  trees  in  all,  there  were  4  times  as  many  cherry  trees  as 
pear  trees,  and  twice  as  many  apple  trees  as  cherry  trees.  How 
many  trees  were  there  of  each  kind  ? 

19.  A  number  plus  itself,  plus  twice  itself,  plus  4  times  itself, 
is  equal  to  72.     What  is  the  number  ? 

20.  Charles  is  twice  as  old  as  his  younger  brother,  and  half  as 
old  as  his  older  brother.  If  the  sum  of  the  ages  of  the  three 
brothers  is  28  years,  what  is  the  age  of  each  ? 

21.  A  farmer  had  twice  as  many  sheep  as  horses,  and  twice  as 
many  hogs  as  sheep  and  horses  together.  If  there  were  in  all 
360  animals,  how  many  were  there  of  each  kind  ? 

22.  A  tract  of  land  containing  640  acres  was  divided  into  three 
farms,  such  that  the  first  was  3  times  as  large  as  the  second,  and 
the  third  4  times  as  large  as  the  first.  How  many  acres  did  each 
farm  contain  ? 

23.  Three  boys  divided  160  marbles  among  themselves  so  that 
one  of  them  received  twice  as  many  as  each  of  the  others.  How 
many  did  each  receive  ? 

24.  Divide  30  into  two  parts,  one  of  which  is  14  times  the 
other. 

25.  Divide  18  into  three  parts,  such  that  the  first  is  twice  the 
third,  and  the  second  is  3  times  the  third. 


10  ACADEMIC  ALGEBRA 

26.  Divide  21  into  three  parts,  such  that  the  first  is  twice  the 
second,  and  the  second  is  twice  the  third. 

27.  Divide  36  into  three  parts,  such  that  the  first  is  twice  the 
second,  and  the  third  is  equal  to  twice  the  sum  of  the  first  and 
second. 

28.  Three  newsboys  sold  60  papers.  If  the  first  sold  twice  as 
many  as  the  second,  and  the  third  sold  3  times  as  many  as  the 
second,  how  many  did  each  sell  ? 

29.  Henry  earned  a  certain  number  of  dollars  per  week.  With 
4  weeks'  earnings  he  purchased  a  rifle,  and  with  20  weeks'  earn- 
ings, a  bicycle.  If  both  together  cost  $  72,  how  much  did  he  earn 
per  week  ?     How  much  did  the  rifle  cost  ?  the  bicycle  ? 

30.  A  man  sold  some  ducks  for  50  cents  each,  and  the  same 
number  of  geese  for  75  cents  each.  If  for  all  he  received  $  12.50, 
how  many  of  each  did  he  sell  ? 

31.  John  has  5  times  as  much  money  as  James.  James  has 
24  cents  less  than  John.     How  much  has  each  ? 

32.  A.  man  had  675  sheep  in  three  fields.  If  there  were  twice 
as  many  in  the  first  field  as  in  the  second,  and  twice  as  many  in 
the  third  field  as  in  both  of  the  others,  how  many  sheep  were 
there  in  each  field  ? 

33.  A  man  bequeathed  to  his  daughter  twice  as  much  money 
as  to  his  son,  and  to  his  wife  3  times  as  much  as  to  his  daughter. 
If  all  together  received  $  9000,  how  much  did  each  receive  ? 

34.  A  plumber  and  two  helpers  together  earned  $  7.50  per 
day.  How  much  did  each  earn  per  day,  if  the  plumber  earned  4 
times  as  much  as  each  helper  ? 

35.  What  number  added  to  |  of  itself  equals  20  ? 

Solution 
Let  X  =  the  number. 

Then,  a;  +  i«  =  20, 

fa;  =  20, 

ia;  =  4. 
Therefore,  x  =  12,  the  number. 


ALGEBRAIC   SOLUTIONS  11 

36.  If  ^  of  a  number  is  added  to  the  number,  the  sum  is  12. 
What  is  the  number  ? 

37.  If  ^  of  a  number  is  added  to  twice  the  number,  the  sum  is 
35.     What  is  the  number  ? 

38.  The  difference  between  4  times  a  certain  number  and  \  of 
the  number  is  30.     What  is  the  number  ? 

39.  The  difference  between  |  of  a  certain  number  and  f  of  it 
is  16.     What  is  the  number  ? 

40.  After  spending  \  of  my  money  and  losing  \  of  it,  I  had 
$  30.     How  much  had  I  at  first  ? 

41.  The  difference  between  twice  a  certain  number  and  J  of 
it  is  20.     What  is  the  number? 

42.  The  number  150  can  be  divided  into  two  parts,  one  of 
which  is  f  of  the  other.     What  are  the  parts  ? 

43.  One  part  of  45  is  |  of  the  other.     What  are  the  parts  ? 

44.  Find  two  parts  of  30  such  that  one  is  \  of  the  other. 

45.  To  A,  B,  and  C  I  owe  in  all  $  93.  If  I  owe  A  |  as  much 
as  C,  and  B  f  as  much  as  C,  how  much  do  I  owe  each  ? 

46.  The  length  of  a  field  is  1|  times  its  width,  and  the  distance 
around  the  field  is  120  rods.  If  the  field  is  rectangular,  what  are 
its  dimensions  ? 

47.  A,  B,  C,  and  D  buy  $  16,000  worth  of  railroad  stock.  How 
much  does  A  take,  if  B  takes  3  times  as  much  as  A,  C  twice 
as  much  as  A  and  B  together,  and  D  ^  as  much  as  A,  B,  and  C 
together  ? 

48.  In  one  season  an  orchard  bore  650  bushels  of  fruit,  con- 
sisting of  J  as  many  bushels  of  pears  as  of  peaches,  and  twice 
as  many  bushels  of  apples  as  of  pears.  How  many  bushels  were 
there  of  each  ? 

49.  A  horse,  harness,  and  carriage  cost  $340.  If  the  horse 
cost  3  times  as  much  as  the  harness,  and  the  carriage  cost  1\ 
times  as  much  as  the  horse,  what  was  the  cost  of  each  ? 


DEFINITIONS   AND   NOTATION 


6.  The  ideas  of  number  and  the  knowledge  of  the  processes 
with  abstract  numbers  that  the  student  has  gained  in  arithmetic 
are  a  proper  and  necessary  introduction  to  his  work  in  algebra ; 
but  since  number  is  discussed  in  a  more  general  way  in  algebra 
than  in  arithmetic,  many  arithmetical  processes,  terms,  and  sym- 
bols, as  ^addition/  'subtraction,'  'greater,'  'less,'  'exponent,'  '  +  ,' 
'  — ,'  etc.,  must  now  be  extended  in  meaning  and  application. 

Por  example,  in  an  arithmetical  sense  8  cannot  be  subtracted 
from  5,  nor  does  8^  have  any  meaning ;  but  in  an  algebraic  sense, 
as  will  be  shown  hereafter,  8  can  be  subtracted  from  5  and  8^  is  as 
intelligible  as  81 

Indeed,  the  processes  and  principles  of  arithmetic  are  but  spe- 
cial cases  of  the  more  fundamental  processes  and  principles  of 
algebra. 

7.  A  unit  or  an  aggregate  of  units  is  called  a  Whole  Number, 
or  an  Integer. 

One  of  the  equal  parts  of  a  unit  or  an  aggregate  of  equal  parts 
of  a  unit  is  called  a  Fractional  Number. 

Such  numbers  are  called  Arithmetical,  or  Absolute  Numbers. 

8.  Arithmetical  numbers  have  fixed  and  known  values,  and  are 
represented  by  symbols  called  numerals;  as  1,  2,  3,  etc.,  Arabic 
figures,  and  I,  V,  X,  etc.,  Eoman  letters. 

9.  It  is  often  convenient,  in  solving  a  problem,  to  employ 
letters,  such  as  x,  y,  z,  to  represent  the  numbers  whose  values  are 
sought ;  and,  in  stating  a  rule,  to  employ  letters  to  represent  the 
numbers  that  must  be  given  whenever  the  rule  is  applied. 

Numbers  represented  by  letters  are  called  Literal  Numbers. 

12 


.      DEFINITIONS  AND  NOTATION  13 

For  example,  the  volume  of  any  rectangular  prism  is  equal  to 

the  area  of  the  base  niultii)lied  by  the  height.     By  using  v  for 

volume,  a  for  area  of  base,  and  h  for  height,  this  rule  is  abbreviated 

to 

V  —  a  X  h. 

When      a  =  60  and  7i  =  5,  v  =  60  x  5  =  300 ; 

when  a  =  36  and  /i  =  10,  v  =  36  x  10  =  360 ;  etc. 

In  each  problem  to  which  this  rule  applies  a  and  h  represent 
fixed,  known  values,  but  in  consequence  of  being  used  for  all 
problems  of  this  class,  a  and  h  represent  numbers  to  which  any 
arithmetical  values  ivhatever  may  be  assigned.  Hence,  the  arith- 
metical idea  of  number  is  extended  as  follows. 

10.  A  literal  number  to  which  any  value  can  be  assigned  at 
pleasure  is  called  a  General  Number. 

11.  A  number  whose  value  is  known  or  a  number  to  which 
any  value  can  be  assigned  is  called  a  Known  Number. 

The  numerals,  3  and  4|,  and  the  general  numbers  a  and  hinv^ax  /i,  in 
§  9,  are  known  numbers. 

Known  literal  numbers  are  generally  represented  by  the  first 
letters  of  the  alphabet. 

12.  A  number  whose  value  is  to  be  found  is  called  an  Unknown 
Number. 

Unknown  numbers  are  usually  represented  by  the  last  letters  of 
the  alphabet. 

ALGEBRAIC  SIGNS 

13.  The  Sign  of  Addition  is  -f,  read  ^ plm.'' 

It  indicates  that  the  number  following  it  is  to  be  added  to  the 
number  preceding  it. 

a  +  &,  read  '  a  plus  6,'  indicates  that  6  is  to  be  added  to  a. 

14.  The  Sign  of  Subtraction  is  — ,  read  '  minus.'' 

It  indicates  that  the  number  following  it  is  to  be  subtracted 
from  the  number  preceding  it. 

a  —  6,  read  '  a  minus  6,'  indicates  that  6  is  to  be  subtracted  from  a. 


14  ACADEMIC  ALGEBRA 

15.  The  Sign  of  Multiplication  is  x  or  •,  read  ^multiplied  hy.^ 

It  indicates  that  the  number  preceding  it  is  to  be  multiplied  by 
the  number  following  it. 

a  X  &,  or  rt-6,  indicates  that  a  is  to  be  multiplied  by  h. 

The  sign  of  multij)lication  is  usually  omitted  in  algebra,  except 
between  figures. 

a  X  &,  or  a-h^  may  be  abbreviated  to  ah.xx  y  to  xy,  4c  x  b  to4  &,  etc.  But 
3x5  cannot  be  written  3o,  because  35  means  30  +  5. 

16.  The  Sign  of  Division  is  -r-,  read  'divided  by.^ 

It  indicates  that  the  number  preceding  it  is  to  be  divided  by 
the  number  following  it. 

«  -f-  6  indicates  that  a  is  to  be  divided  by  b. 

Division  may  be  indicated  also  by  writing  the  dividend  above 
the  divisor  with  a  line  between  them. 

Such  indicated  divisions  are  called  Fractions.     (Cf.  §  158.) 

-  indicates  that  a  is  to  be  divided  by  b. 
b 

17.  The  Sign  of  Equality  is  =,  read  'is  equal  to'  or  'equals.' 

18.  The  Sign  of  Inequality  is  >  or  <. 

When  used  between  two  numbers,  it  signifies  that  they  are 
unequal,  the  greater  number  being  at  the  opening  of  the  sign. 

a  >  6  is  read  '  a  is  greater  than  6.' 
a;  <  5  is  read  '  x  is  less  than  5.' 

19.  The  Signs  of  Aggregation  are :  the  Parenthesis,  (  ) ;  the 
Vinculum,  ;  the  Brackets,  [  ]  ;  the  Braces,  \  \  ;  and  the  Vertical 
Bar,  \. 

They  show  that  the  expressions  included  by  them  are  to  be 
treated  as  single  numbers. 

Thus,  each  of  the  forms  (a  +  &)c,  a  +  b  -c,  [a  +  6]c,  {a  +  6}c,  and  a\c, 
signifies  that  the  sum  of  a  and  b  is  to  be  multiplied  by  c.  +  b\ 

When  numbers  are  included  by  any  of  the  signs  of  aggregation,  they  are 
commonly  said  to  be  in  parenthesis. 

20.  The  Sign  of  Continuation  is  •  •  •  •  or ,  read  '  and  so  on,' 

or  '  and  so  on  to.' 

2,  4,  6,  8, is  read  '  2,  4,  6,  8,  and  so  on.' 

21.  The  Sign  of  Deduction  is  .-..     It  signifies  therefore  or  hence 


DEFINITIONS  AND  NOTATION  16 

FACTORS,  POWERS,  AND  ROOTS 

22.  Each  of  two  or  more  numbers  which  multiplied  together 
produce  a  given  number  is  called  a  Factor  of  the  number. 

Since  12  =  2  x  6,  or  4  x  3,  each  of  these  numbers  is  a  factor  of  12. 
Since  3  a&  =  3  x  a  x  &,  each  of  these  numbers  is  a  factor  of  3  ah. 

23.  When  a  factor  of  a  number  is  considered  as  the  multiplier 
of  the  remaining  factor,  it  is  called  a  Coefficient  of  that  factor. 

In  7x,  5  ax,  hxy^  and  (a  —  &)a;,  the  coefficients  of  x  are  7,  5  a,  6y,  and 
(a  —  &) ;  in  hxy^  hx  is  the  coefficient  of  ij. 

Coefficients  are  Numerical,  Literal,  or  Mixed,  according  as  they 
are  composed  oi  figures,  letters,  ov  hoi\i  figures' and  letters. 

When  no  numerical  coefficient  is  expressed,  the  coefficient  may 
be  considered  to  be  1. 

24.  When  a  number  is  used  a  certain  number  of  times  as  a 
factor,  the  product  is  called  a  Power  of  the  number. 

Powers  are  named  from  the  number  of  times  the  number  is  used 
as  a  factor. 

When  a  is  used  twice  as  a  factor,  the  product  is  the  second  power  of  a ; 
when  a  is  used  three  times  as  a  factor,  the  product  is  the  third  power  of  a  ; 
four  times,  the  fourth  power  of  a  ;  n  times,  that  is,  any  number  of  times,  the 
nth  power  of  a. 

The  second  power  is  also  called  the  square,  and  the  third  power  the  cube. 

The  product  indicated  by  axaxaxaxa  may  be  more  briefly 
indicated  by  a^.  Likewise,  if  a  is  to  be  used  n  times  as  a  factor, 
the  product  may  be  indicated  by  a". 

25.  A  figure  or  letter  placed  a  little  above  and  to  the  right  of 
a  number  is  called  an  Index  or  an  Exponent  of  the  power  thus 
indicated. 

The  integers  that  the  student  has  been  using  in  arithmetic  have  been 
positive  integers. 

When  the  exponent  is  a  positive  integer,  it  indicates  the  num- 
ber of  times  that  the  number  is  to  be  used  as  a  factor. 

52  indicates  that  5  is  to  be  used  twice  as  a  factor ;  a^  indicates  that  a  is  to 
be  used  3  times  as  a  factor. 

When  no  exponent  is  written,  the  exponent  is  regarded  as  1. 
6  is  regarded  as  the  first  power  of  5,  and  a}  is  usually  written  a. 


16  ACADEMIC  ALGEBRA 

The  terms  coefficient  and  exponent  should  be  carefully  distin- 
guished. 

Thus,     5a  =  a  +  a  +  a  +  a  +  «,  but  a^=ivxxaxrtxaxa. 

26.  One  of  the  equal  factors  of  a  number  is  called  a  Root  of  the 
number. 

5  is  a  root  of  25  ;  a  is  a  root  of  a*  ;  4  a;  is  a  root  of  64  x^. 

Roots  are  named  from  the  number  of  equal  factors  into  which 
the  number  is  separated. 

One  of  the  two  equal  factors  of  a  number  is  its  second  root ;  one  of  the 
three  equal  factors  of  a  number  is  its  third  root ;  one  of  the  four  equal  factors, 
the  fourth  root ;  one  of  the  n  equal  factors,  the  nth  root. 

The  second  root  of  a  number  is  also  called  its  square  root,  and  its  third 
root  is  called  its  cube  root. 

27.  The  symbol  which  denotes  that  a  root  of  a  number  is  sought 
is  y',  written  before  the  number. 

It  is  called  the  Root  Sign,  or  the  Radical  Sign. 
A  figure  or  letter  written  in  the  opening  of  the  radical  sign 
indicates  what  root  of  the  number  is  sought. 
It  is  called  the  Index  of  the  root. 
When  no  index  is  written,  the  second,  or  square  root  is  meant. 

VS  indicates  that  the  third,  or  cube  root  of  8  is  sought. 

y/ax  and  Va  —  h  indicate  the  square  roots  of  ax  and  a  —  &,  respectively. 

The  horizontal  line  used  in  connection  with  the  radical  sign  is  a 
vinculum. 

ALGEBRAIC  EXPRESSIONS 

28.  A  number  expressed  by  algebraic  symbols  is  called  an 
Algebraic  Expression. 

29.  AVhen  signs  of  operation  are  employed  in  algebraic  ex- 
pressions, the  sequence  of  operations  is  determined  by  the  follow- 
ing conventional  law : 

A  series  of  additions  and  subtractions  or  of  multiplications  and 
divisions  are  performed  in  order  from  left  to  right. 

3  +  4-2  + .3=    7-2  +  3  =  5  +  3=    8. 
3x4-2x3  =  12 -2x3  =  0x3  =  18. 
a-^-h  —  c  +  d  indicates  that  h  is  to  be  added  to  a,  then  from  this  result  c 
is  to  be  subtracted,  and  to  the  result  just  obtained  d  is  to  be  added. 


DEFINITIONS  AND  NOTATION  17 

30.   When  a  particular  number  takes  the  place  of  a  letter  or 
general  number,  the  process  is  called  Substitution. 

Numerical  Substitutions 

1.    When  a  =  2,  6  =  3,   and  c  =  ^,  what   are    the    numerical 
values  of  3  c,  (?,  VS  a6^,  a?  -f  W,  and  (a  +  6)^,  respectively  ? 

Solutions 
3  c  =  3  .  5  =  15. 

c8  =  5  .  5  .  5  =  126. 


V8a6=^  =  \/8  .  2  .  3  .  3  =  V2. 2x2- 2x3- 3  =  2  x  2  x  3  =  12. 

a2  + 62  =  2. 2  +  3. 3  =  4  +  9  =  13. 

(a  +  6)2  =(a  +  6)(a  +  6)  =  (2  +  3)(2  +  3)=  5  •  5  =  25. 

Find  the  numerical  value  of  each  of  the  following  algebraic 
expressions,  when  a  =  5,  6  =  3,  c  =  10,  m  =  4,  71  =  1 : 


2. 

10  a. 

11. 

{ahf. 

19. 

^4ac^m. 

3. 
4. 

2ab. 
3  cm. 

12. 
13. 

a?h\ 

20. 

c  +  2m 

V2ac/i. 

c-2m 

6. 
6. 

7. 
8. 

6  6c. 
bcrn?. 

2  a26. 

3  6ml 

14. 
15. 
16. 

3  W<m\ 
a?  -  h\ 

(a -by. 

21. 
22. 

0+     2m    , 
c  —  2m 

a'c. 

9. 

3a«6. 

17. 

(n  +  1)*. 

23. 

m" -*. 

10. 

am*. 

18. 

n'  +  1. 

24. 

(6m)-- 

31.  An  algebraic  expression  whose  parts  are  not  separated  by 

-f  or  —  is  called  a  Term ;  as  2ar',  —  5  xyZj  and  ^« 

z 

In  the  expression  2  x2  —  5  xyz  +  ^  there  are  three  terms. 

z 
The  expression  mia  +  6)  is  a  term,  the  parts  being  m  and  (a  +  6). 

32.  Terms  that  contain  the  same  letters  with  the  same  expo- 
nents are  called  Similar  Terms. 

3  x2  and  12  ic2  are  similar  terms  ;  also  3  (a  +  6)2  and  12  (a  +  6)2  ;  also  ax 
and  6a;,  regarding  a  and  6  as  the  coefficients  of  x. 

ACAD.    ALG.  2 


18  ACADEMIC  ALGEBRA 

33.  Terms  that  contain  different  letters,  or  the  same  letters 
with  different  exponents,  are  called  Dissimilar  Terms. 

5  a  and  3  by  are  dissimilar  terms ;  also  3  ofih  and  3  aft^. 

34.  Each  literal  factor  of  a  term  is  called  a  Dimension  of  the 
term. 

The  number  of  literal  factors  or  dimensions  of  a  term  indicates 
its  Degree. 

abed  is  a  term  of  the  fourth  degree,  because  it  is  composed  of  four  literal 
factors,  or  has  four  dimensions,  x^  is  a  term  of  the  third  degree,  since  x^  is 
a  convenient  way  of  indicating  that  x  is  taken  three  times  as  a  factor.  The 
expressions  4:X^yz^  and  5xyz^  are  each  of  the  sixth  degree. 

35.  The  term  of  highest  degree  in  an  expression  determines  the 
Degree  of  the  Expression. 

x^  +  3  a:^  +  .r  +  2  and  abc  ■}■  b-c -\-  ac  —  b  are  expressions  of  the  third  degree. 

36.  When  all  the  terms  of  an  expression  are  of  the  same  degree, 
the  expression  is  called  a  Homogeneous  Expression. 

x^ -^S  x^y -{-xy^ +2  y^  and  abc+b'^c-\-ac'^  are  each  homogeneous  expressions. 

37.  An  algebraic  expression  of  one  term  only  is  called  a 
Monomial,  or  a  Simple  Expression. 

xy  and  oab  are  monomials. 

38.  An  algebraic  expression  of 'more  than  one  term  is  called,  a 
Polynomial,  or  a  Compound  Expression. 

3  a  +  2  6,  xy  +  yz-^  zx,  and  a'^  -\-b'^  —  c^  +  2  ab  are  polynomials. 

39.  A  polynomial  of  two  terms  is  called  a  Binomial. 
3  a  +  2  6  and  x^  —  y'^  are  binomials. 

40.  A  polynomial  of  three  terms  is  called  a  Trinomial. 

a-\-b-\-c  and  ^x  —  2y—z  are  trinomials. 

41.  An  expression,  any  term  of  which  is  a  fraction,  is  called  a 
Fractional  Expression. 

—  3a;  +  -  is  a  fractional  expression. 

42.  An  expression  that  contains  no  fraction  is  called  an  Inte- 
gral Expression. 


DEFINITIONS   AND  NOTATION  19 

5a2_2a  and  Qx  are  integral  expressions. 

Expressions  like  x^  -\-^x^  -\-\x-\-\  are  sometimes  regarded  as  integral, 
since  the  literal  numbers  are  not  in  fractional  form. 

43.  An  expression  that  can  be  written  without  using  a  root 
sign  is  called  a  Rational  Expression. 

1,  2,  3,  •••,  a  4-  i,  -^ — ,  and  (a  —  by  are  rational  expressions. 
x-y 

V25  is  rational,  since  it  can  be  written  5  without  a  root  sign. 

44.  An  expression   that   cannot   be  written  without  using   a 
root  sign  is  called  an  Irrational  Expression. 

a  +  y/b,  a  +  2  Va  +  1,  and  v^4  are  irrational  expressions. 
y/a^  is  not  irrational,  however,  since  it  may  be  written  a. 


POSITIVE    AND    NEGATIVE    NUMBERS 

45.    For  convenience,  arithmetical  numbers  may  be  arranged 
m  an  ascending  scale: 

0,      1,      2,      3,      4,      5,       ... 

I         I         I I  I 


The  operations  of  addition  and  subtraction  are  thus  reduced  to 
counting  along  a  scale  of  numbers.  2  is  added  to  3  by  beginning 
at  3  in  the  scale  and  counting  2  units  in  the  ascending,  or  additive 
direction;  and  consequently,  2  is  subtracted  from  3  by  beginning 
at  3  and  counting  2  units  in  the  descending,  or  subtractive 
direction.  In  the  same  way  3  is  subtracted  from  3.  But  if  we 
attempt  to  subtract  4  from  3,  we  discover  that  the  operation  of 
subtraction  is  restricted  in  arithmetic,  inasmuch  as  a  greater 
number  cannot  be  subtracted  from  a  less.  If  this  restriction 
held  in  algebra,  it  would  be  impossible  to  subtract  one  literal 
number  from  another  without  taking  into  account  their  arith- 
metical values.  Therefore,  this  restriction  must  be  removed  in 
order  to  proceed  with  the  discussion  of  numbers. 

To  subtract  4  from  3  we  begin  at  3  and  count  4  units  in  the 
descending  direction,  arriving  at  1  on  the  opposite,  or  subtractive 
side  of  0.  It  now  becomes  necessary  to  extend  the  scale  1  unit 
in  the  subtractive  direction  from  0. 


20  ACADEMIC  ALGEBRA 

To  subtract  5  from  3  we  begin  at  3  and  count  5  units  in  the 
descending  direction,  arriving  at  2  on  the  opposite,  or  subtractive 
side  of  0.  The  scale  is  again  extended,  and  may  be  extended 
indefinitely  in  the  subtractive  direction  in  a  similar  way. 

For  convenience,  numbers  on  opposite  sides  of  0  are  dis- 
tinguished by  means  of  the  small  signs  +  and  ~,  called  signs  of 
quality,  or  direction  signs,  +  being  prefixed  to  those  which  stand 
in  the  additive  direction  from  0  and  ~  to  those  which  stand  in 
the  subtractive  direction  from  0. 

The  former  are  called  Positive  Numbers,  the  latter  Negative 
Numbers. 

Hence,  the  scale  of  algebraic  numbers  may  be  written  : 

...,      -5,     -4,    -3,     -2,     -1,      0,    n,     +2,     +3,     +4,     +5,        ... 

I         I         I         I         I         I         I         I         I         I         I 

46.  By  repeating  +1  as  a  unit  any  positive  number  may  be 
obtained,  and  by  repeating  ~1  as  a  unit  any  negative  number 
may  be  obtained.  Hence,  positive  numbers  are  measured  by 
the  positive  unit,  +1,  and  negative  numbers  by  the  negative  unit,  ~1, 
or  by  parts  of  these  units. 

47.  If  "•'1  and  ~1,  or  +2  and  "2,  or  any  two  numbers  numerically 
equal  but  opposite  in  quality  are  taken  together,  they  cancel 
each  other.  For  counting  any  number  of  units  from  0  in  either 
direction  and  then  counting  an  equal  number  of  units  from  the 
result  in  the  opposite  direction,  we  arrive  at  0.     Hence, 

Jf  a  positive  and  a  negative  number  are  united  into  one  number, 
any  number  of  units  or  parts  of  units  of  one  cancels  an  equal  number 
of  units  or  parts  of  units  of  the  other. 

48.  Two  concrete  quantities  of  the  same  kind  are  sometimes 
opposed  to  each  other  in  some  sense  so  that,  if  united,  any 
number  of  units  of  one  cancels  an  equal  number  of  units  of  the 
other.  For  convenience,  such  quantities  are  often  distinguished 
as  positive  and  negative. 

If  money  gained  is  positive,  money  lost  is  negative,  for  any  sum  gained 
is  canceled  by  an  equal  sum  lost.  If  a  rise  in  temperature  is  positive,  a  fall 
in  temperature  is  negative.  If  distances  north  or  west  or  upstream  are 
positive,  distances  south  or  east  or  downstream  are  negative. 


ADDITION 


49.  1.  If  a  man  has  10  dollars  in  one  pocket  and  15  dollars  in 
another,  how  much  money  has  he  ? 

2.  If  in  algebra  money  in  hand  is  considered  a  positive  quan- 
tity, indicate  his  financial  condition  algebraically.  What  is  the 
sum  of  10  positive  units  and  15  positive  units,  that  is,  of  +10  and 
+15  ?  of  +4  and  +8  ?  of  +a  and  +6  ? 

3.  If  a  person  owes  one  man  10  dollars  and  another  15  dollars, 
how  much  does  he  owe  both  ?  Indicate  his  financial  condition 
algebraically,  regarding  a  debt  as  a  negative  quantity. 

4.  What  is  the  sum  of  10  negative  units  and  15  negative  units, 
that  is,  of  -10  and  "15  ?   of  "6  and  "14  ?   of  "a  and  "6  ? 

5.  What  sign  has  the  sum  of  two  algebraic  numbers  that  have 
like  signs  ? 

6.  If  a  man  has  25  dollars  and  owes  15  dollars,  how  much  of 
his  money  will  be  required  to  cancel  the  debt?  How  many 
dollars  will  he  have  after  settlement  ? 

7.  What  is  the  result  when  ~15  is  united  with  +25,  that  is, 
what  is  the  algebraic  sum  of  "15  and  +25  ?  of  "20  and  +10  ?  of  +8 
and  -3  ?   of  +6  and  "10  ? 

60.  The  aggregate  value  of  two  or  more  algebraic  numbers  is 
called  their  Algebraic  Sum. 

The  process  of  finding  the  simplest  expression  for  the  algebraic 
sum  of  two  or  more  numbers  is  called  Addition. 

51.  Principles.  —  1.  The  algebraic  sum  of  two  numbers  with 
like  signs  is  equal  to  the  sum  of  their  absolute  values  with  the  com- 
mon sign  prefixed. 

21 


22  ACADEMIC   ALGEBRA 

2.  The  algebraic  sum  of  two  numbers  with  unlike  signs  is  equal 
to  the  difference  between  their  absolute  values  with  the  sign  of  the 
numerically  greater  prefixed. 

By  successive  applications  of  the  above  principles  any  number 
of  numbers  may  be  added. 

Only  similar  terms  can  be  united  into  a  single  term. 

Principle  1  may  be  established  as  follows : 

The  sum  of  5  positive  units  and  3  positive  units  is  evidently  (5  +  3)  posi- 
tive units,  or  8  positive  units  ;  that  is, 

+5 +  +3  ^+(5 +  3)  =+8. 

Similarly,  whatever  absolute  values  a  and  h  represent,  since  a  times  the 
unit  +1  plus  6  times  the  unit  +1  is  equal  to  (a  +  h)  times  the  unit  +1, 
§  46,  +a  +  +b  =  +(a  +  6). 

Again,  the  sum  of  5  negative  units  and  3  negative  units  is  (5  +  3)  negative 
units,  or  8  negative  units  ;  that  is, 

-5  +  -3  =  -(5  +  3)  =  -8. 

Similarly,  whatever  absolute  values  a  and  6  represent,  since  a  times  the 
unit  -1  plus  h  times  the  unit  -1  is  equal  to  {a  +  b)  times  the  unit  -1, 
§  46,  -a  +  -5  =  -{a  +  6). 

Principle  2  may  be  established  as  follows : 

The  sum  of  5  positive  units  and  3  negative  units  is  2  positive  units,  since^ 
§  47,  the  3  negative  units  cancel  3  of  the  positive  units  and  leave  2  positive 
units ;  that  is, 

+5 +  -3  =  +(5 -3)  =+2. 

The  sum  of  5  positive  units  and  7  negative  units  is  2  negative  units,  since, 
§  47,  the  5  positive  units  cancel  5  of  the  negative  units  and  leave  2  negative 
units ;  that  is, 

+5 +  -7  =-(7 -5)  =-2. 

Similarly,  whatever  absolute  values  a  and  6  represent, 
if  a>6,  +a  +  -b  =  +(a-b), 

for,  §  47,  the  b  negative  units  will  cancel  b  of  the  a  positive  units  and  leave 
(a  —  6)  positive  units  ; 

but  if  6  >  a,  +a  +  -b  =  -(6  -  a), 

for,  §  47,  the  a  positive  units  will  cancel  a  of  the  b  negative  units  and  leave 
(6  —  a)  negative  units. 

Examples 


-3  4-  +4  +  -2  +  +8  +  ~9 
+m  -1-  "n,  if  m>n. 


Find  the  value  of 

1.    +7  +  ^3.            3. 

+7  +  -3. 

5 

2.    -7 +  -3.            4. 

-7  +  +3. 

6 

ADDITION  .  23 

52.  To  conform  with  the  ideas  already  presented,  the  terms 
*  greater '  and  '  less '  must  be  interpreted  as  follows  : 

An  algebraic  number  is  increased,  or  made  greater,  when  a  posi- 
tive number  is  added  to  it,  and  decreased,  or  made  less,  when 
a  negative  number  is  added  to  it. 

Since,  by  §  51,  "3  + +1  =  "2,  -2-fn=-l,  "1  + +1  =  0, 
+1  4-  "'"1  =  +2,  etc.,  in  the  scale  of  algebraic  numbers 

...,  -5,  -4,  -3,  -2,  -1,  0,  +1,  +2,  +3,  +4,  +5,  ..., 

each  number  is  greater  than  the  number  on  its  left  and  less  than 
the  number  on  its  right ;  that  is, 

...,  -3<-2,  -2<-l,  -1<0,  0<n,  +K+2,  +2<+3,  .... 

Note.    -3  <  2  may  be  read  '  -3  is  less  than  -2 '  or  '  -2  is  greater  than  -3.' 

Hence,  it  follows  that : 

1.  Any  positive  number  is  greater  than  zero  and  any  negative 
number  is  less  than  zero. 

2.  Of  two  positive  numbers  that  which  has  the  greater  absolute 
value  is  the  greater,  and  of  two  negative  numbers  that  which  has  the 
'£88  absolute  value  is  the  greater-. 

53.  Abbreviated  notation  for  addition. 

Referring  to  the  scale  of  algebraic  numbers,  it  is  evident  that 
adding  positive  units  to  any  number  is  equivalent  to  counting 
them  in  the  positive  direction  from  that  number,  and  adding 
negative  units  to  any  number  is  equivalent  to  counting  them  in 
the  negative  direction  from  that  number.  Hence,  in  addition, 
the  signs  -\-  and  —  denoting  quality  have  primarily  the  same 
meanings  as  the  signs  -f-  and  —  denoting  arithmetical  addition 
and  subtraction.  For  example,  by  the  definition  of  positive  and 
negative  numbers, 

+1  means  0  +  1  and  ~1  means  0  —  1 ; 
also  +5  means  0  -|-  5  and  ~5  means  0  —  5 ;  etc. 

Hence,  in  addition,  but  one  set  of  signs,  -f  and  — ,  is  necessary, 
and  in  finding  the  sum  of  any  given  numbers,  the  signs  -f  and  — 
may  be  regarded  either  as  signs  of  quality  or  as  signs  of  opera- 
tion, though  it  is  commonly  preferable  to  regard  them  as  signs 
of  operation. 


24  .        ACADEMIC  ALGEBRA 

For  brevity,  it   is   customary  to   omit  the   sign   +   before   a 
monomial   or   before   the  first  term  of  a  polynomial.     But  the 
sign  —  cannot  be  omitted. 
+5  +  +3  +  -6  is  written  5  +  3  -  6  ;  -4  +  +8  +  -2  is  written  -4  +  8-2. 

When  there  is  need  of  distinguishing  between  the  signs  of 
quality  +  and  —  and  the  signs  of  operation  -f  and  — ,  the  num- 
bers and  their  signs  of  quality  may  be  inclosed  in  parentheses. 

Thus,  if  a  =  +  5,  6  =  -  3,  and  c  =  -  2,  then  a  +  6  +  c=(+5)  +  (-3) 
+  (-2);  a-  &-c=(+5)-(-3)-(-2);  a6c  =  (+ 5)(- 3)(- 2);  etc. 

54.  A  term  preceded  by  + ,  expressed  or  understood,  is  called 
a  Positive  Term,  and  a  term  preceded  by  — ,  a  Negative  Term. 

Thus,  in  the  polynomial  3a  +  2&  —  5c  the  first  and  second  terms  are 
positive  and  the  third  term  is  negative. 

Examples 
Write  the  following  with  one  set  of  signs : 

1.  +7 +  +8.  4.    +10 +  -2+    -4. 

2.  +6 +  -5.  5.      -6 +  -3 +  +16. 

3.  -3 +  -7.  6.      +8  + +4+    -5. 

55.  1.  How  does  5  +  3  —  2  compare  in  value  with  5  —  2  +  3, 
or  with  3  —  2  +  5,  or  with  -  2  +  3  +  5  ? 

2.  How  does  a  -\-'b  —  c  compare  in  value  with  h  —  c  -{- a,  or 
with  6  +  a  —  c,  or  with  a  —  c-\-h? 

3.  In  what  order  may  numbers  be  added  ? 

Law  of  Order,  or  Commutative  Law  for  Addition.  —  Algebraic  num- 
bers may  be  added  in  any  order. 

The  Law  of  Order  may  be  established  as  follows : 

We  know  from  arithmetic  that  arithmetical  numbers  may  be  added  in  any 
order.  Since,  §  51,  Prin.  1,  algebraic  numbers  having  like  signs  are  added 
by  prefixing  their  common  sign  to  the  sum  of  their  absolute,  or  arithmetical 
values,  and  since  in  finding  this  sum  the  absolute  values  may  be  added  in 
any  order,  it  follows  that  algebraic  numbers  having  like  signs  may  be  added 
in  any  order. 

If  some  of  the  numbers  are  positive  and  some  are  negative,  §  47,  the  same 
number  of  positive  and  negative  units  will  cancel  each  other,  and  the  same 
number  of  one  or  the  other  will  be  left,  in  whatever  order  the  numbers  are 
added. 


7. 

+a  +  -b. 

8. 

-a  +  +6  +  -c. 

9. 

-x  +  -y-{-  -z. 

ADDITION  25 

Hence,  whether  the  numbers  have  like  or  unlike  signs,  they  m^j  be  added 
in  any  order  ;  that  is, 

a  +  6  +  c  =  6  +  c  +  a  =  c  +  a  +  6  =  c4-&  +  a,  etc.^ 
for  all  values  of  a,  &,  and  c. 

66.  1.  How  are  the  numbers  4,  \,  2,  and  f  grouped  \v  adding? 
the  numbers  25  and  32,  or  20  -f-  5  and  30  +  2  ? 

2.  In  what  manner  may  the  terms  of  an  expression  he  cfroupe(? 
in  addition  ? 

Law  of  Grouping,  or  Associative  Law  for  Addition.  —  The  »am  oj 
three  or  more  algebraic  numbers  is  the  same  in  whatever  may^Mfiff  the 
numbers  are  grouped. 

The  Law  of  Grouping  may  be  established  as  follows : 

By  the  Law  of  Order,      a4-6  +  c  =  6  +  c  +  a 
§29,  =(j)j^c)-{-a 

by  the  Law  of  Order,  =  a  +  (6  +  c). 

Other  cases,  as  a  -\-  h  +  c  =  {a  -{■  c)  -\-  h^  etc. ,  may  be  proved  similarly 

Hence,  for  all  values  of  a,  6,  and  c, 

a  +  6  +  c  =  a  4-  (6  +  c)  =  (a  +  c)  +  6  =  c  +  (a  +  6),  etc. 

57.    To  add  similar  monomials. 

Examples 

1.  Add  4  a  and  3  a. 

PROCESS  Explanation. — Just  as  4  =  1  +  1  +  14-1,  so  4a=a+a  +  a+a: 

4  d  just  as  3  =  1  +  1  +  1,  so  3  a  =  a  +  a  +  a.     Therefore,  4  a  +  3  a 

o  =a-\-a  +  a-\-a-Ya-\-a-\-a^  the  symbol  for  which  is  7  a. 

Or,  the  sum  may  be  obtained  by  adding  the  numerical  coefl&- 

7  a  cients  and  annexing  to  their  sum  the  common  literal  part. 

2.  Add  4a,  fa,  —3a,  and  \a. 

PROCESS 

4a+-|a-3a  +  ia  =  4a-3a-|-(fa  +  ia)  =  a  +  2a  =  3a. 

Explanation. — By  the  Law  of  Grouping  the  sum  of  fa  and  \a  may 
be  added  to  the  sum  of  4  a  and  —  3  a.  Just  as  4  =  1  +  1  +  1  +  1,  so 
^a  =  a-{-a-\-a-\-a',  just  as  — 3=— 1  — 1  — 1,  so  — 3a=— a  — a  — a. 
Therefore,  4a  —  3a  =  a  +  a  +  a+a-a-a-a=  (by  the  Law  of  Order) 
a-a+rt-a  +  a-«  +  a  =  0  +  a  =  a.  Just  as  f  =  i  +  i  +  i,  so  fa 
=  \a  +  \a-\-\a.  Therefore,  |a+  \a  =  \a-\-\a  +  \a-\-\a=z\a  —  2a. 
Adding  the  two  groups,  a  +  2a  =  a  +  a  +  a  =  3rt. 

Or,  the  sum  may  be  obtained  by  adding  in  any  order  or  manner  the 
numerical  coefficients,  and  annexing  to  their  sum  the  common  literal  part. 


26  ACADEMIC  ALGEBRA 

Simplify  the  following : 

3.  2y-7y-5y-y  +  10y-6y  +  Sy. 

4.  5a  —  3a  +  8a  —  10 a  —  5a  —  11  a  + 24 a. 

5.  3by -5by- 10 by -Uby  +  iSby. 

6.  Sa^b-\-Qa^b-lla^b-2a^b-\-9a^b. 

7.  If  X^f  -i^^f-  lyV  ^y  +  H  ^f  +  ^f' 

8.  5  (a^?/)2  -  3  (xyy  -  15  (aj?/)^  +  4 (ajy)^  +  13 (a^)^. 

9.  (a  —  x)  +  5  (a  —  x)  4-  7  (a  —  ic)  —  3  (a  —  a;)  —  2  (a  —  x). 

10.  3  (a  4-  ?>)'4-  6(a  +  6)^-  10(a  +  by  -  (a  +  6)2+  12(a  +  bf. 

11.  20  Va;  -  3  -  8  V^^^  -  12  Va;  -  3  +  V^^^+  7  Vx^^. 

12.  3  a;(ar^  -  2  a;  +  3)  -  a;(a;2  -  2 a;  +  3)  +  2  a;(a^  -  2  a;  4-  3). 

13.  2(a;  -  1)  -  13(aj  -  1)  +  5(a;  -  1)  +  10(aj  _  1)  +  6(a;  -  1) 

14.  i(a-\-b-c)-^(a  +  b-c)-\-^(a  +  b-c). 

Since  only  similar  terms  can  be  united  into  a  single  term,  in 
algebra  dissimilar  terms  are  considered  to  have  been  added  when 
they  have  been  written  in  succession  with  their  proper  signs. 

In  algebra  many  indicated  operations  are  regarded  as  per- 
formed. 

Since  5  a,  —  3  &,  and  2  c  cannot  be  united  into  a  single  term,  their  sum 
is  written  5a  —  3&  +  2c. 

15.  Add  6a,  —5b,  —3a,  3b,  2c,  and   —a. 
Solution.     6a-56-3a  +  36  +  2c  —  a  =  2a  —  26-f2c. 

Add  the  following : 

16.  2xy,  4tab,  3xy,  and  ab. 

17.  mn,   —3cd,  —6m7i,  and  4cc?. 

18.  a,   —b,  2  c,  —2  a,  3  b,  and  —4  c. 

19.  6x,  3y,   —2x,  y,   —3x,  z,  and  —3y. 

20.  2a,  2b,  2c,  2d,  -a,  -3b,  -  c,  and  -3d, 

21.  a,  —4a,  2b,  cd,  —2ab,  5b,  and  —3cd. 


ADDITION  27 

58.   To  add  polynomials. 

Examples 

•     1.   Add  3a  — 36  + 5c,  —3 «  + 26,  and  c  — 46  + 2a. 

PROCESS  Explanation.  — For  convenience  in  adding,  simi- 

3a  —  36  +  5c  lar  terms  are  written  in  the  same  column. 

—  3  a  +  2  6  '^^^  algebraic  sum  of  the  first  column  is  2  a,  of 

2a  —  Ah  A-     c  ^^®  second  —  5b,  and  of  the  third  +Qc;  and  these 

numbers  written  in  succession  express  in  its  simplest 

2a  —  56  +  6c  form  the  sum  sought. 

2.  Simplify  11  a26  -  7  a6^  +  2  ac^  + 10  a6  -  4  ac2  + 5  a''6- 4  a62 
+  5ac2  +  6^  +  9a62  -  7  a26  -2  6^  +  2a62  -  8a6  -6a'b. 

PROCESS 

11  a26  -  7  a62  +  2  ac2  +  10  a6  +     6^ 
+  5a26-4a62-4ac2  -26^ 

-7a26  +  9a62  +  5ac2 
-6a26  +  2a62  -  •8a6 

3a26  +3ac2+    2a6-    b^ 

Rule.  —  Arrange  the  terms  so  that  similar  terms  stand  in  the 
same  column. 

Find  the  algebraic  sum  of  each  column,  and  write  the  results  in 
accession  with  their  proper  signs. 

3.  Add  2a-36,  26-3c,  5c-4a,  lOa-56,  and  76-3c. 

4.  Add  x-{-y  -\-Zj  x  —  y-\-Zf  y  —  z  —  x,  z  —  x  —  y,  and  x  —  z. 

Simplify  the  following  polynomials : 

5.  7ic  — II2/  +  42  — 7  2;  +  lla;  — 4y  +  72/— II2;— 4a;+2/— a;— 2;. 

6.  a  +  3  6  +  5c  —  6a  +  d  +  46  —  2c  —  26  +  5a  —  d  +  a  —  6. 

7.  4ar^-3a^  +  52/2  +  10a^-17/-lla^-5a^  +  12a^-2a^. 

8.  2xy-by^  +  s?y'^-7xy-\-3y'^-4.x^y^-{-6xy-\-Ay'^-\-^y\ 

9.  2  ay  —  3  ac  —  4  a^/  +  4  ac  —  6  a2/  +  5  ac  + 11  a^/  —  4  ac  —  a^/. 


28 


ACADEMIC  ALGEBRA 


10.  5  am  — 3  aV  +  4  —  4  am  +  a^m^  —  2  +  5  +  a-m^  —  6  +  3  am. 

11.  6Vx  —  5V^  -f  3Vy  —  4Va;  +  Q^xy  —  V^  —  V2/  +  3V^ 

—  2  V^. 

Add  the  following  polynomials : 

12.  7a-3&  +  5c-10c?,  2  6  +  c?-3c-4e,  5c-6a  +  2(^-4e, 
85_7a  —  8c—  e,  a  —  5c  +  5dH- 11  e,  a  —  6  +  c  +  2d-|-e,  and 
5a-46  +  2c. 

13.  5x-3y-2z,  ^y-2x-\-^z,3a-2x-4.y,4:h-2z  —  by, 
a  — 5  b,  5y  —  6x,  Sx-{-2y  —  5a  —  2b,  and  6x —  y  —  2z -]- 4:b. 

14.  m+n— Vmn,  Vmn— 2m— 3w,  3m+2n,  4n  — Vm/i— 3m, 
5 Vmn  —  yi,  4  m  —  n  —  2Vmn,  5  n  +  2  m  —  3Vmn,  and  n  —  6  m. 

15.  2c  — 7d  +  6»i,  11m  — 3c  — 5 n,  7n  — 2(Z  — 8c,  8cZ  — 3m 
+  10c,  4d  —  3n  —  8m,  m  —  6w,  and  2m  — 3d. 

16.  4aj8-2a^-7a;  +  l,  a^  +  3a^+5a;-6,  4a;2-8a^  +  2-6a;,   ^ 
2(B8-2ar^  +  8a;  +  4,  and  2a^  -  30^  -  2a;  + 1. 

17.  a'  +  5a'b  +  5ab'-{-b',  a'b  -  2  a' -i- ai'b^  -  2  b',  a'b^-3a^b^ 
-4:a*b-a%  and  2  a' +  a*b -2a%^ +  2a^b^ -3ab* +  b'. 

18.  a^-2a36+3a262,  3a63-4  6^-2a262,  3a^b+4.a'-3ab^+4:b\ 
5a^b-^7a^b^-4.ab^-Sb\  and  a*  -  6 a^d  -  8 a^ft^ 

19.  5af-a?-\-7x-9,4.x^-3af-\-6a^-\-12,a^-5x^-x-7y 
4_aJs_jc«,  4a;^-10ar^  +  3a^  +  4,  and  a;«  +  a^- 3ar^- 4a;- 5. 

20.  3(a  +  6)  +  6(6  +  c),  5(a4-&)-10(&  +  c),  2(a  +  6)  +  (6  +  c), 
3(6  +  c)  -  (a  +  6),  2(6  +  c)  -  10(a  +  6),  and  3(a  +  6)  -  3(6  +  c). 

21.  a;  +  3(a  +  l)-2/,  -(a  +  l)-2a;  +  42/,  and  3a;  -  4(a +  1). 

22.  a^-3a^bc-6ab%  o?b  -  b^  -  (?  -  3  abc,  ab^ +  b''c  +  b(?, 
5  a26c  +  4  a62c  +  c^,  6^  -  a'b  -  ab\  a«  +  6^0  +  6c2,  and  2  a62c  -  2  6c2. 

23.  .12a;»-4a;2_^a;+2,  A x" - 4: x -\- .4:  - a^,  3^a;-.6+3a;2^2a;3^ 
and  l-ix  +  1.2x'-\-^x'. 

24.  aa;  —  f  aar*  —  i  ax-^,  |  ao;*  —  ^  aor'  —  J  6a;?/,  i  bxy  —  }  ao;^  —  J  a6, 
1 6a;2/  —  i  «&  +  i  cto;,  and  2  a6  —  |  a»  -f- 1  aa;^. 


ADDITION  29 


Exercises 


69.  1.  If  a  boy  has  n  marbles  and  buys  10,  how  many  will  he 
then  have  ?     If  he  gives  away  m  of  these,  how  many  will  be  left  ? 

2.  Mary  has  25  cents.  How  many  cents  will  she  have  after 
spending  10  cents  and  earning  a  cents  ?  If  she  has  c  cents,  spends 
6  cents,  and  earns  a  cents,  how  many  cents  will  she  have  ? 

3.  A  boy  who  has  p  marbles  loses  q  marbles,  and  then  buys 
r  marbles.     How  many  does  he  then  have  ? 

4.  James  is  15  years  old.  In  how  many  years  will  he  be  21 
years  old  ?  In  how  many  years  will  he  be  x  years  old  ?  Harry 
is  y  years  old.  How  many  years  older  is  he  than  James?  In 
how  many  years  will  Harry  be  x  years  old  ? 

5.  Edith  is  14  years  old.  How  old  was  she  4  years  ago?  a 
years  ago  ?     How  old  will  she  be  3  years  hence  ?  h  years  hence  ? 

6.  William  is  x  years  old.  How  old  was  he  a  year  ago? 
How  old  will  he  be  in  5  years  ?  in  a  years  ?  After  how  many 
years  will  he  be  21  years  old  ?  m  years  old  ?  How  old  will  he 
be  when  he  is  twice  as  old  as  he  is  now  ? 

7.  In  a  certain  family  there  are  five  children  each  of  whom 
is  2  years  older  than  the  one  next  younger.  If  the  youngest  is 
X  years  old,  what  are  the  ages  of  the  others  ? 

8.  A  woman  sold  some  eggs,  and  with  the  money  bought  8 
pounds  of  sugar  and  5  pounds  of  coffee.  If  the  sugar  cost  a 
cents  a  pound,  and  the  coffee  cost  b  cents  a  pound,  how  much  did 
she  receive  for  the  eggs  ? 

9.  What  two  whole  numbers  are  nearest  to  50  ?  to  a;?  to 
a; -1-5?  If  2/  is  an  even  number,  what  are  the  nearest  even 
numbers  ? 

10.  George  is  a  years  younger  than  Henry,  and  h  years 
younger  than  John.     If  John  is  16  years  old,  how  old  is  Henry  ? 

11.  A  man  paid  two  men,  whom  he  owed,  in  the  following 
manner:  To  the  first  he  gave  an  a-dollar  bill,  and  received 
change  amounting  to  h  dollars;  and  to  the  second  he  gave  a  h- 
dollar  bill,  and  received  change  amounting  to  c  dollars.  How 
much  did  he  owe  both? 


30  ACADEMIC  ALGEBRA 

Equations  and  Problems 

60.  1.  Simplify  the  equation  2x  — Sx-\-%x  +  6x  —  x=z21, 
and  find  the  value  of  x. 

Solution 

2a:-3x  +  6x  +  5a;-a;  =  27. 
Uniting  terms,  9  ic  =  27. 

Hence,  x  =  3. 

Simplify,  and  find  the  value  of  a; : 

2.  10  x-7x-^4:X-6x-{-llx- 20  x-^  12x^4.0. 

3.  13x-6x-4:X-^7  x-\-llx-16x-]-lox  =  20. 

4.  25x-5x-7 x-2x-^14:x-10x-12x  =  36. 

5.  17x-^2x-6x-^4:X-12x-30x-\-40x  =  75. 

6.  10  X  -\-2  X  -\-  3  X  -\-  4:  X  +  11  X  +  12  x  +  IS  X  =  60. 

7.  16x-3x-5x-Sx  +  l0x-^15x-15x  =  50. 

8.  12  a;  +  10  aj  -  20  a;  +  16  «  -  3  .T  -  2  a;  4-  2  a;  =  75. 

9.  14  a;  -  11  a;  +  26  a;  -  35  a;  -  4  a;  +  7  a;  +  4  a;  =1 16. 

10.  75  a;  -  37  a;  -  40  a;  +  10  a;  -  8  a;  -  6  .-c  +  9  x  =  21. 

11.  4  a;  +  10  a;  -  60  a;  +  48  a;  +  12  a;  +  5  X  +  2  a;  =  63. 

12.  7  a;  +  11  a;  -  13  a;  +  15  a;  -  17  a;  -  3  a;  +  5  a;  =  25. 

13.  5  a;  -  15  aj  + 25  a; -30  a; +  10  a; +  3  a; +  6  a;  =  56. 

14.  aj  +  2a;  +  3a;  +  4a;-f5a;  +  6aj  +  7a;  +  8a^  =  144. 

15.  4  a;  +  12  ^  -  17  a;  -  10  a;  +  15  a;  -h  a?  +  15  a;  =  400. 

16.  3a;-|-10a;-20a;-4a;4-12a;  +  3a;  +  lla;  =  300. 

Solve  the  following  problems  : 

17.  A  man  bequeathed  $10,000  to  3  sons  and  4  daughters,  so 
that  a  son  received  twice  as  much  as  a  daughter.  What  was  the 
share  of  each  daughter,  and  of  each  son  ? 


ADDITION  31 

18.  John  had  twice  as  many  marbles  as  Henry,  and  J  as  many 
as  Charles.     If  they  had  225  marbles  in  all,  how  many  had  each  ? 

19.  A  had  twice  as  much  money  as  B,  who  had  3  times  as 
much  money  as  C.  If  all  together  had  $  2000,  how  much  money 
had  each? 

20.  A  merchant  owes  A  a  certain  sum  of  money,  B  i  as  much, 
and  0  twice  as  much  as  A.  Various  persons  owe  him  in  all  12 
times  as  much  as  he  owes  B.  If  all  these  debts  were  paid,  he 
would  have  ^  10,000.     What  are  the  amounts  he  owes  ? 

21.  Mr.  Jones  succeeded  in  doubling  his  capital  once  every 

5  years.     If  his  capital  at  the  end  of  20  years  was  $  150,000,  with 
what  capital  did  he  begin  ? 

22.  The  distance  around  a  rectangular  field  4  times  as  long  as 
it  is  wide  is  200  rods.     What  are  the  dimensions  of  the  field  ? 

23.  What  are  the  dimensions  of  a  rectangular  field  whose 
length  is  twice  its  width,  if  240  rods  of  fence  are  required  to 
inclose  it? 

24.  Two  boys  caught  the  same  number  of  fish,  another  caught 
10  more,  and  another  10  less.  If  they  caught  in  all  120  fish,  how 
many  did  each  catch  ? 

25.  The  sum  of  3  consecutive  whole  numbers  is  84.  What 
are  the  numbers? 

Suggestion.  —  Let  x  represent  the  middle  number.  Then,  the  other  two 
numbers  will  be  represented  by  x  —  1  and  a;  +  1. 

26.  Of  what  3  consecutive  even  numbers  is  150  the  sum  ? 

27.  A,  B,  C,  and  D  together  have  $  1500.  If  A  had  $  50  more 
and  B  $  50  less,  they  would  each  have  the  same  sum  as  C  and  ^ 
as  much  as  D.     How  much  money  has  each  ? 

28.  The  ages  of  4  brothers  differ  successively  by  2  years.  If 
the  sum  of  their  ages  is  56  years,  what  is  the  age  of  each  ? 

29.  Three  newsboys  sold  270  papers  in  an  evening.  If  the 
second  sold  5  less  than  twice  as  many  as  the  first,  and  the  third 

6  more  than  3  times  as  many  as  the  first,  how  many  papers  did 
each  sell? 


SUBTRACTION 


61.  1.  What  is  left  when  5flJ2/  is  taken  from  12  xy?  What  is 
the  sum  of  "o  xy  and  +12  xy  ? 

2.  What  is  left  when  +3  mn  is  subtracted  from  +10  mn  ?  What 
is  the  sum  of  ~3  mn  and  +10  mn  ? 

3.  Instead  of  subtracting  a  positive  number,  what  may  be 
done  to  obtain  the  same  result  ? 

4.  What  is  the  result  when  8  a  is  subtracted  from  10  a  ?  When 
(Sa—  5a)  is  subtracted  from  10 a ?  How  does  the  second  result 
compare  with  the  first  ?  What  effect  upon  the  result  has  the 
subtraction  of  the  negative  number  ~5a? 

5.  How  does  the  result  of  subtracting  (5x  —  2x)  from  12  a; 
compare  with  the  result  of  subtracting  5  x  from  12  a;  ?  What 
effect  upon  the  result  has  the  subtraction  of  the  negative  number 
-2  a;? 

6.  Instead  of  subtracting  a  negative  number,  what  may  be 
done  to  obtain  the  same  result? 

62.  In  addition  two  numbers  are  given,  and  their  algebraic 
sum  is  required;  in  subtraction  the  algebraic  sum,  called  the 
minuend,  and  one  of  the  numbers,  called  the  subtrahend,  are 
given,  and  the  other  number,  called  the  remainder,  or  difference, 
is  required. 

Subtraction  is,  therefore,  the  inverse  of  addition. 
The  Difference  is  the  algebraic  number  that  added  to  the  sub- 
trahend gives  the  minuend. 

63.  Principles.  —  1.  Subtracting  a  positive  number  is  equiva- 
lent to  adding  a  numerically  equal  negative  number. 

2.  Subtracting  a  negative  number  is  equivalent  to  adding  a  numer- 
ically equal  positive  number. 


SUBTRACTION  33 

The  difference  of  similar  terms,  only,  can  be  expressed  in  one 
term. 

Principle  1  may  be  established  as  follows : 

Let  m  represent  any  minuend  and  +s  any  positive  subtrahend. 

It  is  to  be  proved  that  w  —  +s  =  m  +  -s. 

§  62,  to  find  the  remainder  when  +s  is  subtracted  from  m  is  to  find  the 
algebraic  number  that  added  to  +s  will  give  m. 

Since  the  algebraic  sum  of  +s  and  -s  is  0,  by  the  Associative  Law  for 
Addition  the  algebraic  sum  of  +s  and  m  +  -s  is  w  +  0,  or  w. 

Hence,  the  algebraic  number  that  added  to  +s  gives  m  is  m  +~s. 

.'.  m  —+s  =  in  -\-~s. 

Principle  2  may  be  established,  as  follows : 

Let  m  represent  any  minuend  and  s  any  negative  subtrahend. 

It  is  to  be  proved  that  m  —-s  =  w  ++s. 

§  62,  to  find  the  remainder  when  ~s  is  subtracted  from  m  is  to  find  the 
algebraic  number  that  added  to  s  will  give  m. 

Since  the  algebraic  sum  of  s  and  +s  is  0,  by  the  Associative  Law  for 
Addition  the  algebraic  sum  of  s  and  m  ++s  is  w  +  0,  or  m. 

Hence,  the  algebraic  number  that  added  to  s  gives  m  is  m  ++s. 

.'.  m  —s  =  m  -\--^s. 

64.  Since,  from  the  above  principles,  subtracting  algebraic 
numbers  is  equivalent  to  adding  them  to  the  minuend  with  their 
signs  changed,  it  follows  that  the  Laws  of  Order  and  Grouping 
for  Addition  hold  in  the  subtraction  of  algebraic  numbers ;  and 
that  when  one  or  more  subtrahends  with  their  signs  changed  are 
added  to  the  minuend  to  form  the  algebraic  sum  called  the  dif- 
ference, one  set  of  signs,  +  and  — ,  suffices  to  denote  either  quality 
or  operation. 

65.  To  subtract  when  the  terms  are  positive. 

Examples 
1.     Prom  10  a;  subtract  4  ». 


4a; 


PROCESS 

^Q  jp  Explanation.  —  Since  subtracting  a  positive  term  is  equiv- 

alent to  adding  a  numerically  equal  negative  term  (Prin.  1), 
4  X  may  be  subtracted  from  10  x  by  changing  the  sign  of  4  x, 
and  adding  10  x  and  —  4  x. 

ILG.  — 3 


34 


ACADEMIC  ALGEBRA 


2.    From  10  a?  subtract  15  a;. 


PROCESS 

10  a; 
15  a; 

—  5x 


From 
Take 


From 
Take 


From 
Take 


Explanation.  —  Since  subtracting  a  positive  terra  is 
equivalent  to  adding  a  numerically  equal  negative  term 
(Prin.  1),  15 X  may  be  subtracted  from  10 x  by  changing  the 
sign  of  15x  and  adding  10  x  and  —  15x. 


12  a 

5a 


4. 

9  am 
21  am 


9. 

9a  +  76 
2a  +  36 

13. 

15  m  +    n 
\2m-\-2n 


5. 

10. 

5a +  105 

7a+    46 

14: 

lx^2y 

4:X  -\-  4?/ 


6.  7.  8. 

24  mn^        6^  ax      11  (a  +  b) 
12  mn^      15Vax      21(a  +  b) 


11. 

10x-\-2y 
6x  -^  Ay 

15. 

4a;  -f  4?/ 

7x-{-2y 


12. 

Sm  +  3n 
2m  -{-  5n 

16. 

10p  +  2g 


17.  From  ^p  +  3z  subtract  lOp-^-z. 

18.  From  15  m  +  n  subtract  5  m  +  3  ti. 

19.  From  '6ax-{-5hy  subtract  4  aa;  +  6  by. 

20.  From  8  a&c  + 19  ma;  subtract  20  a?>c  + 7  ma?. 

21.  From  a  +  .3  6  +  c  subtract  a  +  ?>  +  3c. 

22.  From  12 a^  +  2 b"' ^  14. c"  subtract  3a-  + 13  6^  +  3  01 

23.  From  6  aa*  +  % -j- 7  C2;  subtract  2  ax -{- by -{- 2  cz. 

24.  From  1  ax -\- by  -\- 2  cz  subtract  4:ax-\-oby  +  cz. 

25.  From  4a6  +  c  subtract  a^ -{- b^ -j- abc -{- 2  ab -]- 2  c. 

26.  From  5xy  subtract  a;^  4.  2  a;2^  +  2  a;/ +  3  a;^/ +  2/^. 

27.  From  1  subtract  a;^  + 13  a;^  + 15  a;^  _^  ^g  ^  _^  25. 

28.  From  7a;2-4/  subtract  6x^-\-3xy-6y\ 


SUBTRACTION  35 

66.   To  subtract  when  some  terms  are  negative. 

Examples 

1.    From  8x  — 3?/  subtract  5x  —  7y. 

PROCESS  Explanation.  —  Since  subtracting  a  positive  temi  is  equiv- 

g  3- 3  ^        alent  to  adding  a  numerically  equal  negative  term,  subtract- 

K     rr  ing  5  X  from  8  x  is  equivalent  to  adding  —5  a;  to  8  .r  (Prin.  1). 

Since  subtracting  a  negative  term  is  equivalent  to  adding 

H a  numerically  equal  positive  term,  subtracting  —  7  y  from 

3a;-|_42/        —Sy  is  equivalent  to   adding    +ly  to   —3?/    (Prin.  2). 

Rule.  —  Change  the  sign  of  each  term  of  the  subtrahend,  or  con- 
ceive it  to  be  changed,  and  add  the  result  to  the  minuend. 


From 
Take 

2.               3. 

5a           6ocy 
-2a        -Sxy 

4.                     5. 

—  9mn        —  13  V^ 

—  4m?i        —    oVo^ 

6. 

-    S(a  +  b) 
-10{a-hb) 

From 
Take 

7. 
4:m  —  3n  +  2p 
2  7?i  —  5  n  —    p 

8. 

8a-106  +  c 
6a—    ob  —  c 

9. 

Sx-\-2y-z 
5x  —  4y  —  z 

From 
Take 

10. 

a  —  b-\-c 
2a-\-b-c 

11. 

Sa^b  —  5a<^  +  9a^c 
3  a-b  +  2  ac^  -  9  a^c 

12. 

r  —  s  +  t 
r-irs  —  t 

13.  From  5x  —  Sy-\-z  take  2x  —  y-\-^z. 

14.  From  3  a'b  +  W  -  a^  take  4  a-6  -  8  a^  +  2  b\ 

15.  From  13  a^  +  5  b-  -  4  c^  take  8  a^  +  9  6^  +  lo  c\ 

16.  From  \5x  —  Sy-\-2z  subtract  3  a; +  8  2/  — 92. 

17.  From  a'-ab-b^  subtract  ab-2a^-2b\ 

18.  From  m^  —  mn-\-n^  subtract  2  m' —  Smn-\-2n\ 

19.  From  oQiy^  —  2xy  —  y^  subtract  2  x^  -{-2xy  —  3y\ 

20.  From  2ax  —  by  —  5xy  subtract  2by  —  2ax  —  3xy. 


86  ACADEMIC  ALGEBRA 

21.  From  2a-\-c  subtract  a  —  b-\-c. 

22.  From  2m-{-n  subtract  n  —  2 p. 

23.  From  x-\-y  subtract  3  a  —  4  -(-  ?/. 

24.  From  2x^-{-2xy  subtract  x^  —  xy  —  y\ 

25.  From  2a  — 2d  subtract  a  —  b-\-c  —  d. 

26.  From  2  b  subtract  b  —  a  —  c  —  d. 

27.  From  a^  +  cc^  subtract  a^  —  3  a^x -{- 3  aa^  —  a?. 

28.  From  a*  + 1  subtract  1  —  a  +  a^  —  a^  +  a"*. 

29.  From  the  sum  of  3o?  —  2ab  —  W  and  3ab  —  2a'^  subtract 
a^-ab-b\ 

30.  From    3x  —  y-\-z   subtract   the    sum   of    a;  — 4?/ +  2    and 
2x  +  3y-2z. 

31.  From   a-\-b-\-c  subtract  the  sum  of  a  —  b  —  c,  b  —  c  —  a, 
and  c  —  a  —  b. 

32.  Subtract  the  sum  of  7in?n  —  2  mn^  and  2  m-n—m^—n^-{-2mn^ 
from  m^  —  n^. 

33.  Subtract  the  sum  of  2c  —  9a  —  36  and  36  —  5a  —  5c  from 
6  —  3  c  +  a. 

34.  From    3  6a; +  4  a?/   subtract  the   sum   of    3ay  —  4:bx    and 
bx  -\-  ay. 

85.    From  the  sum  of  1  +  a;  and  1—x^  subtract  1  —  x -\- a^  —  x^. 

36.  From  ^a:^- ^x' -\-3x-7  subtract  ^x^  -  lx^-\- ^x-10. 

37.  From  ^  m^  —  ^  mhi  +  i  m?i-  —  -^j  n^  subtract  n^  —  m^  +  J  wi^^^ 

38.  From    5  (a  +  6)  —  3  (a;  +  2/)  +  4  (m  +  n)    subtract    4  (a  +  6) 
'  +  2(a;  +  2/)  +  (m  +  n). 

39.  From    n^  —  m^    subtract   the   sum   of    2mhi^  —  3mn^  and 
m^  +  4  m^w^  —  2  m^/i^  +  5  mn^  —  n-\ 

40.  From  the  sum  of  3a^  —  2x-\-l  and  2a;  — 5  subtract  the 
sum  of  a;  —  a;^  H- 1  and  2  a;^  —  4  a;  4-  3. 


SUBTRACTION  37 

PARENTHESES 

67.  The  subtrahend  is  sometimes  written  within  a  sign  of 
aggregation  preceded  by  the  sign  — . 

If  a  —  6  is  to  be  subtracted  from  2  a,  it  may  be  written  2  a  —  (a  —  6). 

1.  What  change  must  be  made  in  the  signs  of  the  terms  of  the 
subtrahend,  when  it  is  subtracted  from  the  minuend  ? 

2.  When  a  number  in  parenthesis  is  preceded  by  the  sign  — , 
what  change  must  be  made  in  the  signs  of  the  terms,  when  the 
subtraction  is  performed,  or  when  the  paj'enthesis  is  removed  f 

68.  Principles.  —  1.  A  parenthesis  preceded  by  the  minus  sign 
may  be  removed  from  an  expression,  if  the  signs  of  oil  the  terms  in 
parenthesis  are  changed. 

2.  A  parenthesis  preceded  by  the  minus  sign  may  be  used  to  inclose 
an  expression,  if  the  signs  of  all  the  tei-ms  to  be  inclosed  are  chariged. 

1.  When  numbers  are  inclosed  in  a  parenthesis  preceded  by  the  plus 
sign,  the  parenthesis  may  be  removed  without  changing  the  signs  of  the 
terms. 

2.  Any  number  of  terms  may  be  inclosed  in  a  parenthesis  preceded  by  a 
plus  sign  without  changing  the  signs  of  the  terms. 

3.  The  student  should  remember  that  in  an  expression  like  —  («  —  y),  or 
—  X  —  y,  the  sign  of  x  is  plus,  and  the  expression  is  the  same  as  if  it  were 
written  -  (+  x  -  y),  or  —  +  x  ^  y. 

Examples 
Simplify  the  following : 

1.  a  +  (6  — c).  S.  a  —  b  —  (c  —  d). 

2.  a  —  {b  —  c).  9.  a  — 6  — (— c-l-a). 

3.  X  —  {y  —  z).  10.  a  —  m  —  (n  —  m). 

4.  x-(-y-\-z).  11.  5a-2b-{a-2b). 

5.  m  —  n  —  (—a).  12.  a  —  (b  —  c-\- a) —(c  —  b), 

6.  m-(n-2a).  13.  2  xy -\- S  f  -  (x^ -\- xy  -  f). 

7.  5x  —  {2x-{-y).  14.  m  +  (3  m  —  n)  —  (2  w  —  m)  -f  w. 


38  ACADEMIC  ALGEBRA 

Collect  in  alphabetical  order  the  coefficients  of  x  and  y  in  the 
following,  giving  each  parenthesis  the  sign  of  the  first  coefficient 
to  be  inclosed  therein  : 

15.   ax— by —  3bx-{-2cy —  fx  — gy. 

PROCESS 

ax  —  by—Sbx  +  2cy  —fx  —  gy=(a  —  Sb  —f)x  —  (b  —  2 c  4- g)y 

Suggestion. — The  coefficient  of  y  is  —  6  +  2  c  —  gf,  which  is  written 
-(h-'Zc  +  g)   (Prin.  2). 

16.  ax  —  by  —  bx  —  cy  -\-  dx  —  ey.  21 .  bx  —  cy  —  2ay-\-  by. 

17.  mx  —  2ny-\-  nx  —  ry  —px  -{-  qy.  22.  mx  —  bx  —  'iy  —  my. 

18.  5  ax  -\-  3  ay  —  2  dx  -\-  ny  —  5  X  —  y.  23.  rx  —  ay  —  sx  -{-  2  cy. 

19.  cx  —  2bx-\-7ay-{-Sax  —  lx  —  ty.  24.  x^ -\- ax  —  y^ -\- ay. 

20.  bx-\-  cy  —  2ax  -\-by  —  ex  —  dy.  25.  x^  —  ay  —  ax  —  y"^. 

Group  the  same  powers  of  x  in  the  following : 

26.  aa?  4-  b:)?  —  ex  -{-  ex^  —  dx^  —  fx. 

27.  a^-\-3x^-^3x-ax^-8aa^-]-bx. 

28.  a^  —  abx  —  y?  —  bar  —  ex  —  mnx^  -f  dx. 

29.  ax'^  —  x'^  —  ax^  -i-  x^  -^  ax  —  x  —  abx'  +  a^. 

30.  Simplify  2  a-la-\b  -  (3b  -  2a-b)\;\-(b  -  a). 

When  an  expression  contains  parentheses  within  parentheses, 
they  may  be  removed  in  sueeession,  beginning  with  either  the 
outermost  or  the  innermost,  preferably  the  innermost. 

Solution 


2a-la-{b-(Sh-2a-  6)}]  -(b-a) 

Prin.  1,  =2a-la-{b-(3b-2a  +  b)};\-b-\-a 

Uniting  terms,  =  3  a  —  [a  —  {&  —  (4  &  —  2  a)}]  —  b 

Prin.  1,  =  3a -[a -{6 -46  + 2a}]- 6 

Uniting  terms,  =  3  a  —  [a  —  {—  3  6  +  2  a}]  —  6 

Prin.  1,  =3a-[a  +  3  6-2a]-6 

Uniting  terms,  =3a— [— a  +  3&]— 6 

Prin.  1,  =3a  +  a-36-6 

Uniting  terms,  =  4  a  —  4  6. 


SUBTRACTION  39 

Simplify  the  following : 

31.  ^a^-b  —  \x  +  ^a-\-h  —  2y  —  (x-\-y)], 

32.  ah  —  \ab  -\-  ac  —  a  —  (2  a  —  ac)  -\-  {2  a  —  2 ac)]. 

33.  a-f-[y-55+4a-(62/  +  3)J-(72/-4a-l)]. 

34.  4  m  —  [p  +  3 n  —  (m  4-  n)  +  3  —  (6i?  —  3  n  —  5  m)]. 


35.  a  +  26  +  (14a -56) -{6a +  66 -(5a- 4a -46)J. 

36.  12a-J[4-36-(664-3c)]+6-8-(5a-26-6)i. 

37.  a+6—  \  — [a4-&-(c+ic)]-[3a— (c— a;+a)-6]+4a}. 

38.  aj3  _[^^  _  (1  _  a,)]  _  jl  +[aj2_  (1  _a;)  ^_  a^j  j. 


39.  4  -  J[5^y  -  (3  -  2a;  -  2)]  -  [a;  +(5 ?/  -  a;  +  3)]}. 

40.  a6  —  {5  +  a;  —  (6  4-  c  —  a6  +  a;) S  +  [a;  —  (6  —  c  —  7)]. 

41.  d'-W-  \ad  -h  a"  -  (a;  +  a2  -  6^)  -  6^1  +  5ad  -  (a;  +  3ad). 

42.  a-(6-c)-[a-J6-c-(6+c-a)  +  (a-6)  +  (c-a)S]. 

43.  —  J3aa;  — [5a;y-32]  -f  2  —  (4a;?/ +[6z -f- 7aa;]-f  30)J. 

44.  1  -x-fl  —  a;-[l  —  x  —  (1  —  .t)  —  (x  —  1)]- a;  + If. 

45.  1  -  a;  —  Jl  —  [a;  —  1  +  (a;  —  1)  —  (1  —  .t)  —  X']  +  1  —  a;j. 

46.  .'c-[-J-(-a;)  +  a;5  -2a;].      . 

47.  (a-6)-j-a-(6-a)-|-(a-6)S. 


48.  a-7-[-  5-a-(-a-a-3)J]. 

49.  a  —  a;  — [— [a  +  (a;  — a)  —  (a;  — 4a)|]. 


50.    ^xy-\_-\{y'^  —  xy)-{xy-y^-2xy)\'\. 


51.    2a-[a- J6-(36-2a-6)n-(&-a)- 


52.  a— [— (m  —  a)  —  ja  — (m  —  2m  +  6a)j]. 

53.  a  — 1-6-  (c-cf)J  +  a-[-6  +  J-2c-(d-e)J]. 


54 .    a2  +  5  -  [2  a6  -  J  -  (7  -  3  a6)  -  a6  +  2  a=^  -  2  S  -  (3  a  -  z) ]. 


55.    2a;+(32/-S2a;-[?/H-4a;-(32/-a^)]-22/S-a;-2/). 
66.    l-(-J-[--.(-a-^iri)-3]-2;-a)-[a-(a-l)]. 


40  ACADEMIC  ALGEBRA 

TRANSPOSITION  IN  EQUATIONS 

69.    1.  What  number  diminished  by  2  is  equal  to  8  ? 

2.  If  a  number  increased  by  2  is  equal  to  8,  what  is  the 
number  ? 

3.  In  the  equation  a;  —  2  =  8,  what  is  done  with  the  2  in 
obtaining  the  value  oi  x?  In  the  equation  a;  =  8  -f  2,  how  does 
the  sign  of  2  compare  with  its  sign  in  the  previous  equation  ? 

4.  In  the  equation  x  -\-2  =±S,  what  is  done  with  the  2  in 
obtaining  the  value  of  x  ?  In  the  equation  a;  =  8  —  2,  how  does 
the  sign  of  2  compare  with  its  sign  in  the  previous  equation  ? 

5.  In  changing  the  2's  from  one  side  of  the  equation  to  the 
other,  what  change  was  made  in  the  sign  ? 

6.  When  a  term  is  changed  from  one  side,  or  member  of  an 
equation,  to  the  other,  what  change  must  be  made  in  its  sign  ? 

7.  If  3  is  added  to  one  member  of  the  equation  2  +  5  =  7, 
what  must  be  done  to  preserve  the  equality? 

8.  If  3  is  subtracted  from  one  member  of  the  equation 
2  +  8  =  10,  what  must  be  done  to  preserve  the  equality  ? 

9.  If  one  member  of  the  equation  2  +  5  =  7  is  multiplied  by 

5,  what  must  be  done  to  preserve  the  equality  ? 

10.  If  one  member  of  the  equation  10  -f-  25  =  35  is  divided  by 

6,  what  must  be  done  to  preserve  the  equality  ? 

11.  If  one  member  of  the  equation  a;  =  5  is  raised  to  the  second 
power,  what  must  be  done  to  preserve  the  equality  ? 

12.  If  the  square  root  of  one  member  of  the  equation  x^  =  25 
is  taken,  what  must  be  done  to  preserve  the  equality  ? 

13.  What,  then,  may  be  done  to  the  members  of  an  equation 
without  destroying  the  equality  ? 

70.  The  parts  of  an  equation  on  each  side  of  the  sign  of 
equality  are  called  its  Members, 

The  part  on  the  left  of  the  sign  of  equality  is  called  the  First 
Member,  and  the  part  on  the  right,  the  Second  Member. 


SUBTRACTION  41 

71.  The  process  of  changing  a  term  from  one  member  of  an 
equation  to  the  other  is  called  Transposition. 

72.  Principle.  —  A  term  may  he  transposed  from  one  member 
of  an  equation  to  the  other,  provided  its  sign  is  changed. 

73.  A  truth  that  does  not  need   demonstration  is  called  an 
Axiom. 

74.  Axioms.  —  1.    Things  that  are  equal  to  the  same  thing  are 
equal  to  each  other. 

2.  If  equals  are  added  to  equals^  the'sxims  are  equal. 

3.  If  equals  are  subtracted  from  equals,  the  remainders  are  equal. 

4.  If  equals  are  multiplied  by  equals,  the  products  are  equal. 

5.  If  equals  are  divided  by  equals,  the  quotients  are  equal. 

6.  The  same  powers  of  equal  numbers  are  equal. 

7.  The  same  roots  of  equal  numbers  are  equal. 

Equations  and  Problems 

75.  1.    If  5  a;  -  2  =  3  iB  +  6,  find  the  value  of  x. 


PROCESS  Explanation.  — Since  the  value  of  x  is  sought, 

K     2  =  SiC-k6      ^^^  terms  containing  x  must  be  collected  in  one 

member  of  the  equation,  and  the  remaining,  or 
^nown  terms  in  the  other  member. 

3  a;  may  be  made  to  disappear  from  the  second 


Sx         =Sx 


2a?-2=       +6 

2  =  2 member  by  subtracting  3  x  from  both  members. 

2  X  =8  The  result  (Axiom  3)  is  the  equation  2  x  —  2  =  6. 

.'.  a;  =  4  ~  2  °^^y  he  made  to  disappear  from  the  first 

member  by  adding  2  to  both  members.     The  result 
(Axiom  2)  is  the  equation  2  x  =  8. 
OX  — 2     =3a;-h6  Dividing  both  members  of  this  equation  by  2, 

5a/'  —  3a;  =  2+6         the  coefficient  of  x,  the  resulting  equation  (Axiom 
2  a;  =  8  6)  is  x  =  4,  the  value  of  x  sought. 

.  *.  a;  =  4  ^^'  since  a  term  may  be  transposed  from  one 

member  to  the  other  if  its  sign  is  changed  (Prin.), 
VERIFICATION  3  x  transposed  to  the  first  member  becomes  -  3  x 

20  _  2  z=  12  4-  6       ^°^  ~  "  transposed  to  the  second  member  becomes 
+  2.     Therefore,  the  resulting  equation  is 
18  =  18  .        ^        „      « 

5x-3x  =  24-6. 

Uniting  terms,  2  x  =  8.     Dividing  both  members  by  2,  the  result  is  x  =  4. 


42  ACADEMIC  ALGEMA 

The  result  is  verified  by  substituting  the  value  of  x  for  x  in  the  original 
equation.  If  the  members  are  then  identical^  the  value  found  for  the  un- 
known number  is  correct. 

2.   If  5  a;  -  7  =  30  -  7,  find  the  value  oi  x. 

PROCESS 

5  a;  -  7  -  30  -  7. 
bx  =  30. 
.-.  x  =  Q. 

Suggestion.  —  Since,  if  —  7  were  transposed  from  the  first  member  to  the 
second,  it  would  appear  as  +  7  and  cancel  the  term  —  7  in  that  member,  the 
two  equal  terms  may  be  canceled  before  the  transposition. 

Rule.  —  Transpose  terms  so  that  the  unJcnotvn  terms  stand  in  the 
first  member  of  the  equation  and  the  Tcnovm  terms  in  the  second. 

Unite  similar  terms,  and  divide  both  members  of  the  equation  by 
the  coefficient  of  the  unknown  number. 

Verification.  —  Substitute  in  the  original  equation  the  value  of 
the  unknown  number  thus  found.  If  the  members  of  the  equation 
are  then  identical,  the  value  of  the  unknown  number  found  is  correct. 

1.  The  same  term  with  the  same  sign  in  both  members  of  an  equation 
may  be  canceled  (Ax.  2  or  3). 

2.  If  the  signs  of  all  the  terms  of  an  equation  are  changed,  the  equality 
will  not  be  destroyed ;  for  (Ax.  3)  both  members  may  be  subtracted  from 
0  without  destroying  the  equality. 

Find  the  value  of  x  and  verify : 

3.  5a;+3  =  8.  10.  7  x  -  10  =  60. 

4.  x  +  5  =  ll.  11.  7  +  2a^  =  ll. 

5.  a;- 5  =  11.  12.  3  + 2a;  =  15. 

6.  2  a;- 3  =  21.  13.  1 -f  12  a;  =  85. 

7.  5a;  +  7  =  42.  '  14.  5  +  3a;  =  11. 

8.  3a;- 2  =  25.  15.  7  + 5a;  =  47. 

9.  2a;  +  4  =  10.  16.  2  +  9  a;  =  74. 


SUBTRACTTON  48 

17.  3  =  5-0;.  26.  7x-12  =  x  +  13-\-5. 

18.  9-5  a;  =  -1.  27.  4:X -20  =  5  x  -  50 -^  x. 

19.  7x-{-2  =  x-\-14:.  28.  S  x +  16  =  20  -  5  x -\-4:. 

20.  5x-5  =  2x-j-4:.  29.  7  x  -  55  =  18 -2  x-1. 

21.  3x  +  2  =  a;  +  30.  30.  -  a;  -  12  =  40  -  8  a;  +  4. 

22.  5a;-2  =  2aj  +  7.  31.  80  -  3  a;  =  83  -  Sx  +  7. 

23.  2-|-13a;  =  50-9.  32.  9  x  -  90  =  16  -  a; +  4. 

24.  10  +  a;  =  18  — «.  33.  50  — a;  =  20  + a;. 

25.  2a;  +  2  =  32-a;.  34.  7  a;  +  25  =  30  +  6  a;- 3. 

Solve  the  following  problems : 

35.  What  number  increased  by  10  is  equal  to  19  ? 

36.  What  number  diminished  by  30  is  equal  to  20  ? 

37.  What  number  diminished  by  111  is  equal  to  —  15? 

38.  What  number  exceeds  ^  of  itself  by  10  ? 
Suggestion.  —  Let  3  a;  =  the  number. 

39.  Five  times  a  number  exceeds  3  times  the  number  by  14. 

What  is  the  number  ? 

40.  If  5  is  subtracted  from  a  certain  number,  and  the  differ- 
ence is  subtracted  from  3  times  the  number,  the  result  is  35. 
What  is  the  number  ? 

41.  The  double  of  a  number  is  64  less  than  10  times  the  num- 
ber.    What  is  the  number  ? 

42.  If  4  is  subtracted  from  a  certain  number,  and  the  differ- 
ence is  subtracted  from  40,  the  result  is  3  times  the  number. 
What  is  the  number  ? 

43.  Three  times  a  certain  number  is  as  much  less  than  72  as 
4  times  the  number  exceeds  12.     What  is  the  number  ? 

44.  Twice  a  certain  number  exceeds  J  of  the  number  as  much 
as  6  times  the  number  exceeds  65.     What  is  the  number  ? 

45.  If  16  is  added  to  a  certain  number,  the  result  is  56  dimin- 
ished by  7  times  the  number.     AVhat  is  the  number  ? 


44  ACADEMIC  ALGEBRA 

46.  If  6  times  a  certain  number  lacks  as  much  of  62  as  3 
times  the  number  exceeds  19,  what  is  the  number  ? 

47.  Three  times  a  certain  number  increased  by  a  is  equal  to 
the  number  increased  by  9  a.     What  is  the  number  ? 

48.  The  sum  of  4  numbers  in  a  row  is  58.  If  their  common 
difference  is  3,  what  are  the  numbers  ? 

Suggestion.  —  Let  x  =  the  smallest  number. 

Then,  ac  +  3  =  the  second  number, 

jc  +  6  =  the  third  number, 
and  a;  +  9  =  the  fourth  number. 

49.  A  man  distributed  1  dollar  among  5  boys  so  that  each  boy 
except  the  youngest  received  5  cents  more  than  the  boy  next 
younger.  If  the  boys  were  all  of  different  ages,  how  much  did 
each  receive  ? 

50.  The  common  difference  of  5  numbers  is  2,  and  their  sum 
is  100.    What  are  the  numbers  ? 

51.  John  and  James  were  comparing  their  earnings.  John 
said,  "  I  have  earned  50  cents."  James  replied,  "  If  I  had  earned 
half  as  much  as  I  have,  and  10  cents  more,  I  should  have  earned 
the  same  as  you."     How  much  had  James  earned  ? 

52.  A  drover, when  asked  how  many  cattle  he  had,  replied,  "If 
I  had  1  more  than  I  have  and  2  more,  I  should  have  200."  How 
many  cattle  had  he  ? 

53.  The  earnings  of  a  mill  for  4  years  were  $46,000.  If  the 
books  showed  an  annual  increase  of  $  1000,  what  were  the  earn- 
ings for  each  year  ? 

54.  James  had  J  as  much  money  as  John,  John  5  cents  less 
than  William,  and  Robert  5  cents  more  than  3  times  as  much  as 
James.     If  they  together  had  $  1.50,  how  much  had  each  ? 

Suggestion.  — Let  3  x  =  the  number  of  cents  John  had. 

55.  A  speculator  who  doubled  his  money  by  a  fortunate  invest- 
ment, afterward  lost  $  600,  but  he  still  had  $  400  more  than  the 
original  sum.     How  much  had  he  at  first  ? 


MULTIPLICATION 


76.    1.    How  many  are  3ic  +  3aj  +  3a;  +  3ic  +  3a;? 

2.  How  many  are  5  times  3  3;?     2  times  3  a;?     7  times  2a? 

3.  A  man  saves  $  10  a  month,  indicated  by  +10.  How  many 
dollars  will  he  save  in  a  year  ?  What  sign  should  be  placed  be- 
fore the  result  to  indicate  the  number  of  dollars  saved  ? 

4.  How  many  are  12  times  +10  ?  12  times  +10  a  ?  5  times 
+3  a;  ?     8  times  +2  m  ? 

5.  When  a  positive  number  is  multiplied  by  a  positive  num- 
ber, what  is  the  sign  of  the  product  ? 

6.  If  a  man  loses  $5  sl  month,  indicated  by  "5,  how  many 
dollars  will  he  lose  in  a  year  ?  What  sign  should  be  placed  be- 
fore the  result  to  indicate  the  number  of  dollars  lost? 

7.  How  many  are  12  times  ~5  ?  12  times  "5  6?  7  a  times 
-3  a;?     3  times  "2  6?     11  times  Sy?     5  times  "5  a6  ? 

8.  When  a  negative  number  is  multiplied  by  a  positive  num- 
ber, what  is  the  sign  of  the  product  ? 

9.  If  a  man's  gains  in  business  are  $10  a  month,  indicated 
by  +10,  how  many  dollars  less  had  he  3  months  ago,  indicated  by 
~3,  than  he  has  now  ?     Indicate  the  result  algebraically. 

10.  How  many  are  +10  multiplied  by  "3  ?  +10  multiplied  by 
-5  ?     +2  multiplied  by  "3  ?     +a  multiplied  by  ~b  ? 

11.  What  is  the  sign  of  the  product  when  a  positive  number 
is  multiplied  by  a  negative  number  ?  When  a  negative  number 
is  multiplied  by  a  positive  number  ? 

12.  What,  then,  is  the  sign  of  the  product  of  two  numbers 
having  unlike  signs  ? 

45 


46  ACADEMIC   ALGEBRA 

13.  If  a  man  who  is  in  debt  is  getting  deeper  in  debt  at  the 
rate  of  $  10  a  month,  indicated  by  "10,  how  much  better  off  was 
he  3  months  ago,  indicated  by  ~3,  than  he  is  now  ?  Indicate  the 
result  algebraically. 

14.  How  many  are  ""10  multiplied  by  ~3  ?  ~10  multiplied  by 
-5  ?     -2  multiplied  by  "3  ?     'a  multiplied  by  'h  ? 

15.  What  is  the  sign  of  the  product  of  two  negative  numbers  ? 
of  two  positive  numbers  ? 

16.  What,  then,  is  the  sign  of  the  product  of  two  numbers 
having  like  signs  ? 

17.  What  is  the  sign  of  "5  x  "2  ?  of  "5  x  "2  x  "2  ?  of  "5  x  "2 
X+2X-2?  of  +5x-2x-2x-2x-2?  of  "2  x-3  x+2  x+2  x  "2 
X-2X-2? 

18.  What  sign  has  a  product,  if  the  number  of  negative  fac- 
tors is  even9  What  sign  has  a  product,  if  the  number  of  negative 
factors  is  odd'^ 

19.  What,  then,  determines  the  sign  of  a  product? 

20.  In  the  expression  a^,  what  is  3  called  ?  What  does  it  indi- 
cate ?     In  a~  how  many  times  is  a  used  as  a  factor  ? 

21.  When  o?  is  multiplied  by  a"*,  how  many  times  is  a  used  as 
a  factor  in  the  product  ?  when  a^  is  multiplied  by  a^'? 

When  a"'  is  multiplied  by  a",  what  is  the  product,  if  m  and  n 
are  positive  integers  ? 

22.  How,  then,  is  the  exponent  of  a  factor  in  the  product  de- 
termined ? 

77.  When  the  multiplier  is  a  positive  integer,  the  process  of 
taking  the  multiplicand  additively  as  many  times  as  there  are 
units  in  the  multiplier  is  called  Multiplication. 

When  the  multiplier  is  any  number,  multiplication  may  be 
defined  as  the  process  of  finding  a  number  that  has  the  same  rela- 
tion to  the  multiplicand  as  the  multiplier  has  to  1. 

The  multiplicand  and  multiplier  are  called  the  factors  of  the 
product. 

78.  Principles.  —  1.  Law  of  Signs.  —  The  sign  of  the  product 
of  two  factors  is  +  when  they  have  like  signs,  and  —  when  they 
have  unlike  signs. 


MUL  TI PLICA  TION  47 

2.  Law  of  Coefficients.  —  The  coefficient  of  the  product  is  equal  to 
the  x>Toduct  of  the  coefficients  of  the  factors. 

3.  Law  of  Exponents.  —  The  exponent  of  a  number  in  the  j^rod- 
uct  is  equal  to  the  sum  of  its  exponents  in  the  factors. 

79.   The  Law  of  Signs  may  be  established  as  follows : 

+3=+l++l+  +  l,  (1) 

and  -3=-l4.-l+-l=_+l_+l  _+l;  (2) 

that  is,  +3  is  obtained  from  +1  by  taking  +1  additively  three  times,  and    3 
by  taking  +1  subtractively  three  times. 

Hence,  §  77,  multiplying  any  number  by  +3  is  equivalent  to  taking  that 
number  additively  three  times,  and  multiplying  any  number  by  -3  is  equiva- 
lent to  taking  that  number  subtractively  three  times. 

By  (1),  +6  x+3=+5++5++5=+15,  (3) 

and  -5x+3=-5+-5+-5=-15.  (4) 

By  (2),  +5  X -3  =  -+5- +6 -+5= -16,  (5) 

and  -5x-3  =  --5--5--5=  +  15.  (6) 

Similarly,  8ince  +  (§)  =  +  a)  +  +  a)++(i)and-(t)=-+(i)-  +  (D-+(D, 

+5x+(D=+(i)++a)++(D=+(¥);  (7) 

and  so  on,  as  in  (4),  (5),  and  (6). 

In  (3)  and  (6)  the  product  of  two  algebraic  numbers  with  like  signs  is 
positive. 

In  (4)  and  (5)  the  product  of  two  algebraic  numbers  with  unlike  signs  is 
negative. 

(7)  shows  that  like  results  are  obtained  when  the  multiplier  is  a  fractional 
number  (§7). 

Passing  to  general  symbols,  let  a  and  h  be  any  absolute  numbers. 
First,  when  b  is  a  whole  number. 


Since 

+6=+l++l  ++1  +  ... 

•  to  b  terms. 

§77, 

+a  x+b  =+a  ++a  ++a  +  •• 

.  to  b  terms 

=+ab, 

(8) 

and 

-a  x+b  =-a  +~a  +-«  +  •• 

•  to  b  terms 

=  -ab. 

(9) 

Since 

-b  =  -  +  \  -  +  1  -  +  1  - 

•••  to  6  terms, 

§77, 

+a  x-b  =  -+a  -+a  -+a  - 

•  ..  to  b  terms 

=—ab, 

(lo: 

and 

-a  x~b  —  —-a  —~a  —-a  — 

•  ••  to  6  terms 

rr-^nb. 

(ii: 

48  ACADEMIC  ALGEBRA 

Second,  when  h  is  a  fractional  number. 

As  in  (7),  the  same  reasoning  applies  wlien  &  is  a  fractional  number. 

Hence,  from  (8)  and  (11),  the  product  of  any  two  algebraic  numbers  with 
like  signs  is  positive  ;  and  from  (9)  and  (10),  the  product  of  any  two  algebraic 
numbers  with  unlike  signs  is  negative. 

When  the  multiplier  is  a  positive  or  negative  whole  or  fractional 
number,  it  appears  from  the  above  proofs  that  algebraic  multi 
plication  is  only  abbreviated  algebraic  addition.  Hence,  as  in 
addition,  but  one  set  of  signs  +  and  —  is  required  to  denote  botlj 
quality  and  operation. 

Hence,  the  Law  of  Signs  may  be  expressed  as  follows : 

+  a  multiplied  by  +  5  =  +  a5, 

—  a  multiplied  by  —  h  =  +  ah, 

—  a  multiplied  by  -\-h  =—  ah, 
and                        +  a  multiplied  by  —  h  =  —  ah. 

80.  It  follows  from  the  Law  of  Signs,  applied  repeatedly,  that 
the  product  of  any  numher  of  algehraic  numbers  is  +  when  the 
numher  of  negative  factors  is  even,  and  —  when  the  numher  of 
negative  factors  is  odd. 

81.  The  Law  of  Exponents  or  the  Index  Law  for  multiplication 
may  be  established  for  positive  integral  exponents  as  follows : 

Let  m  and  n  be  any  positive  integers. 
By  the  definition  of  a  power,  §  24, 

a"^  =  axaxa...  to  m  factors, 
a^  =  a  X  a  X  a  ..^.  to  n  factors  ; 
.*.  a*"  X  a"  =  (a  X  a  X  a  ...  to  TO  factors)  (a  x  a  X  a ...  to  w  factors) 
=  a  X  a  X  a . ..  to  (m  +  n)  factors. 
Hence,  a"*xa^=  a"»+«. 

In  like  manner,  a^  x  a^  x  ap  =  a"»+«+p. 

Thus,  a2  X  a*  =  a^+*  =  a^, 

and  a^  X  a^  X  a'^  =  a^+^+^  =  aK 

82.  1.  How  does  2x5  compare  with  5  x  2  in  value?  3x7 
with  7x3?     2x5x6  with  2x6x5? 

2.  What  is  the  effect  upon  the  value  of  a  product  of  changing 
the  order  of  its  factors  ? 


MUL  TIPLICA  TION  49 

Law  of  Order,  or  Commutative  Law  for  Multiplication.  —  The  fao- 
tors  of  a  product  may  he  taken  in  any  order. 

The  Law  of  Order  may  be  established  as  follows  : 

Since  the  number  of  negative  factoi-s  will  not  be  changed  by  taking  the 
factors  in  any  order,  §  80,  the  sign  of  the  product  is  the  same  in  whatever 
order  the  factors  are  taken. 

We  know  from  arithmetic  that  arithmetical  numbers  may  be  multiplied 
in  any  order.  Hence,  the  absolute  value  of  the  product  is  the  same  in  what- 
ever order  the  factors  are  taken. 

Since  neither  the  sign  nor  the  absolute  value  of  the  product  of  algebraic 
numbers  is  changed  by  changing  the  order  of  the  factors,  the  factors  may  be 
taken  in  any  order. 

In  general  symbols,  ax?)XcX"-  =  6xcxax---=  etc. 

83.  1.  How  does  2  x  3  x  5,  or  6  x  5,  compare  in  value  with 
2  X  (3  X  5),  or  2  X  15 ?  with  6  xQ'^  a  x  b  x  c,  ov  (ab)  x  c, 
with  a  X  (be)  ? 

2.    How  may  the  factors  of  a  product  be  grouped  ? 

Law  of  Grouping,  or  Associative  Law  for  Multiplication.  —  Tlie  fac- 
tors of  a  product  may  be  grouped  in  any  manner. 

The  Law  of  Grouping  may  be  established  as  follows  : 

By  the  notation  of  multiplication,  abc  denotes  that  a  is  to  be  multiplied 
by  b  and  then  the  product  ab  is  to  be  multiplied  by  c  ;  that  is, 

abc=(ab)c.  (1) 

1.  Let  it  be  required  to  prove  that  (^ab)c  =  a(bc).  . 
By  the  Law  of  Order,  abc  =  bca 

by  notation,  =  (6c) a 

by  the  Law  of  Order,  =  a(bc).  (2) 

From  (1)  and  (2),  (ab)c  =  a(bc). 

Similarly,  it  may  be  proved  that      (ab)c  =  b(ac),  etc. 

2.  Let  it  be  required  to  prove  that  abed  =  (a6)  (cd). 
By  notation,  abed  =  ab  x  c  x  d 

putting  m  for  a&,  =  m  -  c  •  d,  or  mcd. 

By  1,  m-C'd  =  m(cd). 

Putting  ab  for  w,  (ab)  •  c-d  =  (ab)  (cd) ; 

that  is,  by  notation,  abed  =(ab)(cd). 

Similarly,  it  can  be  shown  that  (abe)d  —  a(bcd)  =  (be)  (^ad)  =  (ac) (^bd)  ~ 
(abd)c  ={adc)b  =  e(dba)=  etc.,  the  factors  being  grouped  in  any  manner 
whatever. 

ACAD.    ALG. 4 


50  ACADEMIC  ALGEBRA 

3.  In  a  similar  way  the  law  may  be  established  for  any  number  of  factors, 
successively  for  5,  6,  7,  ...  factors. 

Hence,  abc  "-p  =  a{bc  •••p)=  b(ac  •••p),  etc.,  for  all  values  of  the  letters. 

84.   To  multiply  a  monomial  by  a  monomial. 

Examples 
1.   Multiply  5  icy  by  —  3  xyh. 

Explanation.  —  Since  the  multiplier  is  composed  of  the 

'  factors  —  3,  x,  y^,  and  z,  the  multiplicand  may  be  multiplied 

^   ^1  by  each  successively.     —  3  times  5  x-y^  =  —  15  x:h/'^  (Prin.  1 

—  ^  ^y^        and  2)  ;  x  times   -  15  a^V  =  -  15  x^   (Prin.   3)  ;   y^  times 

—  15  o^t/z        ~  1^  ^^y^  =  —  15  x^y^  (Prin.  3)  ;  and  this  multiplied  by  z  is 

equal  to  —  l^x^y^z  (Prin.  3). 
Or,  since  the  signs  of  the  numbers  are  unlike,  the  sign  of  their  product 
is  —  ;  the  coefficient  of  the  product  is  the  product  of  the  coefficients  5  and  3  ; 
and  the  product  of  the  literal  numbers  is  expressed  by  affecting  each  with 
an  exponent  equal  to  the  sum  of  its  exponents  in  the  factors. 

Rule.  —  To  the  product  of  the  numerical  coefficients  annex  the 
letters,  each  with  an  exponent  equal  to  the  sum  of  its  exponents  in 
both  factors. 

Write  the  sign  +  before  the  product  when  its  factors  have  like 
signs,  and  —  when  they  have  unlike  signs. 


2. 

3.                     4. 

5. 

6. 

Multiply 

-2 

6               -7 

2a 

2m^ 

By 

8 

-2               -9 

5 

^m'^ 

7. 

8.                 9. 

10. 

11. 

Multiply 

10  a' 

x^y^          —  4  abc 

5a'bc^ 

-2xf 

By 

5a^ 

xf               2  a^b 

■  7  abh 

2^y 

12. 

13.                  14. 

15. 

16. 

Multiply 

-Sa'x" 

-Bm^n"    -6a'b'c'x 

4  abed 

-  3  x'by^ 

By 

-2aa? 
17. 

3  mn       —  4  a^bny^ 

-1 

-  1 

18.                 19. 

20. 

21. 

Multiply 

-2aV 

—  3n^y      4  a^xbY 

-1       - 

5  m^HY 

By 

-4  ax* 

6  b^y      3  a^x'b^y 
23.                  24. 

-1       - 
25. 

2m'''n^cY 

22. 

26. 

Multiply 

^pq^^ 

10  mV    -2a2mV 

a^yz 

-pHY 

By 

-2rq^x 

-    3n2m*         Sb'nV 

-x^yz' 

—  abc 

MULTIPLICATION  51 


Multiply 

By 

27. 

2a'»+^ 
3  a' 

32. 

^y 

28. 

—  5  a;" 

X 

33. 

—  a"* 

—  a" 

29. 

—  x'Y 
3xy 

34. 

a" 
38. 

i-i 

30. 

-  x^-y-' 

-xy 

31. 

-  2  a:"--* 

Multiply 

By 

35. 

-x" 

36. 

Multiply 

By 

37. 

Cfn^n-3y2 

39. 

mf'rfbHf 
mhi'^b'y^-'' 

85.  How  does  25  x  2  compare  in  value  with  20  x  2  plus 
5x2?  How  is  133,  or  100  +  30+3,  multiplied  by  2  ?  How  is 
the  polynomial  a  -{-  b  -\-  c  multiplied  by  the  monomial  m? 

Distributive  Law  for  Multiplication.  —  Tlie  product  of  a  polynomial 
by  a  7nono7nkd  is  equal  to  the  algebraic  sum  of  the  partial  products 
formed  by  midtiplying  each  term  of  the  polynomial  by  the  monomial. 

The  Distributive  Law  may  be  established  as  follows : 

Let  a  +  ?;  be  the  multiplicand  and  m  the  multiplier,  a,  6,  and  m  being 
positive  or  negative  integral  or  fractional  numbers. 

By  the  Law  of  Order  the  multiplier  may  change  places  with  the  multi- 
plicand.    Hence,   (a  +  6)  x  w  may  be  written  m{a  +  h). 

It  is  to  be  proved  that  rii{a  +  6)  =  ma  +  mh. 

First,  when  m  is  a  positive  integer. 

Since  m  =  1  4-  1  +  1  +  •••  to  m  terms, 

§  77,     m(a  +  6)  =  (a  +  &)  +  (a  +  6)  +  (a  +  6)  +  ••  •  to  m  terms 
§  50,  =  (a  +  a  +  a  +  •••  to  »n  terms)  +  (6  +  6  +  6  +  •••  to  w  terms) 

§  77,  =  ma  +  mb.  (1) 

Second,  when  m  is  a  fractional  number. 

P 
Let  m  =  — ,  in  which  p  and  q  are  absolute  integers. 

—  (a  +  6)  =  j9  times  one  gth  of  (a  +  6) 
=  p{a  +  6)  gths 
by  (1),  =  pa  gths  +  ph  qths 

=  |of«  +  |ol6 

=  |a  +  |..  (2) 


62  ACADEMIC  ALGEBRA 

Third,  when  m  is  negative. 

Let  m  =  —  w,  w  being  any  positive  whole  or  fractional  number. 
It  is  to  be  proved  that  (—  n) {a  -\-  b)  =  —  7ia  —  nb. 

By  (1)  and  (2),  n(a  +  6)=  wa  +  nb.  (3) 

Since,  if  +  n  is  positive,   —(—  n)  is  also  positive,  substituting  —  (—  w) 
for +  «  in  (3),  _(_„)(,  +  ,)^_(_  „)„_(_„), 

=  —  (—  na  —  nb).  (4) 

Since  both  —(—  7i)(a  +  b)  and   —(—  na  —  nb)  are   now  monomial  in 
form,  both  members  of  (4)  may  be  multiplied  by  the  monomial  —  1. 

.-.  +(^— n)(a  +  b)  =  -\-(— na  — nb), 
or  {—  n)  (a  +  b)  =  —  na  —  nb.  '  (6) 

By  (1),  (2),  and  (5),        m(a  +  b)=  ma  +  mb 
for  all  positive  or  negative  whole  or  fractional  values  of  m  and  for  all  values 
of  a  and  b. 

86.    To  multiply  a  polynomial  by  a  monomial. 

Examples 

1.    Multiply  3a^-/  by   -4?/. 

PROCESS  Explanation.  —  By  §  85,  each  term  of  the  multiplicand 

q  ^ 2  is  to  be  multiplied  by  the  multiplier. 

The  product  of  3  x'^  and  —  4y  is  —  12  x'^y.     But  since 

y the  entire  multiplicand  is  S  x^  —  y"^,  —4y  times  y^  must  be 

12  a^y  4-  4  w^    subtracted  from  —  12  x^y.     —  4  y  times  y"^  =  —  4  y^,  which 

subtracted  from  —  12  x^y  gives  —  12  x^y  +  4  i/. 
Or,  since  a  polynomial  multiplied  by  a  monomial  is  equal  to  the  algebraic 
sum  of  the  partial  products  formed  by  multiplying  each  term  of  the  poly- 
nomial by  the  monomial,  §  85,  Sx^  —  y^  multiplied  by   -  4  ?/  is  equal   to 
-  12  a:%  +  4  ?/3. 

Rule.  —  Multiply  each  term  of  the  polynomial  by  the  monomial, 
and  find  the  algebraic  sum  of  the  partial  products. 


2. 

Multiply    2a2-2a64-362 
By                                    3a6 

3.                               4. 

5  m^  —  4  91^          ^0?  —  2xy  —  y"^ 
—  2  m?n                            —  x-y 

Multiply : 

5.  3x^-2xy  by. 5a;/. 

6.  m"n^~3mn^  by  2mw. 

7.  3a-^-6a26  by  -26. 

8.  pY-2pq''  by   -  pq. 

9.  4a2-5  62c  by  abc\ 

10.   —  2  ac  -f-  4  aa;  by  —  5  acx. 

MUL  TIPLICA  TION  53 

Perform  the  multiplications  indicated : 

1 1 .  d'bc  (3  a^  -  4  a^h  -  5  cv'b'^  +  2  a^^  _  iq  fj4y 

12.  2xy(5a:^  -lOxy -36y' -  5x -\-  r)y  +  120). 

13.  5  m^  (16  7n^  -  20  mhi  + 13  miv^  -  25  n^). 

14.  abc  (a^b^  -  2  aV  _  2  6V  _  a*  _  4  6^  -  c*  -  5  abc). 

15.  -  6c(6^  +  c*  _  ^>s  _  c^  +  ?;2c2  _  4  jj2f.  ^Sbc'-2 be). 

16.  —  2  a;  («*  —  5  a^?/  —  16  x-y-  +  24  .t?/^  —  y*  —  xy  —  x -{-  4). 

87.    To  multiply  a  polynomial  by  a  polynomial. 

To  multiply  p  -\-q  +  r  by  a  +  6, 
§  85,  (p  +  q-h  r) (a  +  h)  =.p(a-\-b)  +  (/(a+  &)  +  r (a  +  &) 

§§  82,  85,  =  aj?  +  6jj>  +  «</  +  ^5  H-  ar  +  6r 

§  55,  =  ap-\-  aq  -{-  ar  -\- bp -{•  bq -\-  br. 

Rule.  —  Multiply  every  term  of  the  multiplicand  by  each  term  oj 
the  multiplier,  and  find  the  algebraic  sum  of  the  2^artial  products. 

Examples 

1.  Multiply  7?  —  xy  by  2x-\-Zy. 

PROCESS 

a?  —  xy 
^x  +3y 

■\-2x  times  {a^  —  xy)  =  2  a:^  —  2  a?y 

+  3  2/  times  {x^  —  xy)  =  +3  ar^?/  —  3  xy^ 

.'.  (2  .T  -|-  3  y)  times  (ar^  —  ayy)  =  2oi^  -\-    a^y  —  3  ocy^ 

2.  Multiply  a,*^  —  3  a^y  -\-  3  xy^  —  y^  by  or  —  xy. 

PROCESS 

a:S_3a^2/  +  3a;/  -^/^ 
0?  —  xy 


ar'  —  3  a;*^  +  3  a^t/^  —     a?'if' 
—    x^y  -^-3  Q^y^  —  3  a^^/^  +  xfy^ 

a^  —  4  a;'*2/  -f-  6  ary  —  4  aj^^  -|-  xy^ 


54 


ACADEMIC  ALGEBRA 


3.    Multiply  ox^a:^-2a^-x'  hj  Sx-]-l 


PROCESS 

TEST 

5x-    2a;2_^^_a.4 

=  +  3 

1     -i-    3x  -x" 

=  +  3 

5x-    2\x'-\-    1 

a^^-t 

X* 

15 

-    6 

+  3 

-3a^ 

-    5 

+  2 

^1 

-\-x' 

5x-{-lSa^-10x^-{-4:X'^-4:X^-^x^ 


+  9 


Suggestion.  —  For  convenience  in  writing  partial  products,  both  polyno- 
mials are  arranged  so  that  in  passing  from  left  to  right  the  several  powers  of 
X  are  either  successively  higher  or  lower.  In  this  process,  the  polynomials 
are  arranged  according  to  the  ascending  powers  of  x. 

Test.  —  Since  the  correct  product  of  6x  ~  2x^  -\-  x^  —  x^  and  1  +  3  aj  —  x^ 
is  the  same  in  form  whatever  value  x  represents,  it  is  possible,  by  assigning 
an  arithmetical  value  to  x,  to  change  the  process  of  multiplying  one  algebraic 
expression  by  the  other  mto  a  process  of  multiplying  one  arithmetical  num- 
ber by  another  as  shown  in  the  test. 

Let  1  be  substituted  for  x. 


Multiplicand  =  5  x  —  2  x^  -f  x^ 
Multiplier      =  1  +  Sx  —  x^ 


24-1 


=  1  +  8 


l=+3 

=  +3 


Product  should  be  equal  to  +9 

5  x  +  13  T2  -  10  a:3  +  4  a:*  -  4  x5  +  x6  =  5  +  18  -  10  +  4  -  4  +  1  =  +  9. 

In  like  manner  the  multiplication  of  any  two  literal  expressions  may  be 
tested  arithnietically  by  assigning  any  values  we  please  to  the  letters.  It  is 
usually  most  convenient  to  substitute  +  1  for  each  letter,  since  this  may  be 
readily  don-'',  by  adding  the  numerical  coefficients. 

Multiply,  and  test  each  result : 
4.    2a;  +  3bya;  +  2.  12.    4.y  -  6b  hy  2y -]-b. 


5.  4.x +  1  by  3a; +  4. 

6.  5w  —  1  by  4?i-f-5. 

7.  h-\-2k  by  3h-k. 

8.  3r-6sby5r-2s. 

9.  4r-\-2s  hj  2r  +  9s. 

10.  3l-{-5t  hj  2l-^6t. 

11.  4a  +  3a;  by  4a  — 3a;. 


13.  3  a;  — 2  2/  by  3  a; +  2  2/. 

14.  2b  +  5chy  5b-2c. 

15.  7a;  — 2nby4a;  +  2n. 

16.  ab  —  15  by  ab  + 10. 

17.  ax  H-  by  by  ax  —  by. 

18.  a^  —  ay  -\- y'^  by  a  -\-y. 

19.  3a2-6a6+3  6-  by  2a-3  6. 


MUL  TIPLTCA  TION  65 

Multiply : 

20.  2oJ'-Sh^-ah  by  S  a" -4.ab -6h\ 

21.  bx-^x'-^lO  by  12-30a;  +  2«2 

22.  3n^  +  3m^  +  mn  by  m^  —  2  m?i^  +  m^n. 

23.  4^-10  +  22/  by  2/-32/  +  5. 

24.  4a;-3a:2-f  2a^  by  3ic-10a^  +  10. 

25.  a^  +  a^  +  4  a^  —  a^  +  a  by  a  +  1. 

26.  31a^-27a:2_^9a;_3  by  3a;  +  l. 

27.  4  ic^  —  3  a:^?/  +  5  a;?/''  —  6?/^  by  5a;H-Gy. 

28.  a  4- &  +  c  +  fZ  by  a  —  6  —  c  4- d. 

29.  a^ -\- h^ -\- c?  —  ah  —  ac  —  he  by  a  +  6  +  c. 

30.  aar^"  +  a?/2"  by  aa^^  —  aif". 

31.  0^"-^  +  ?/""^  by  3  aa;"-^  +  2  2/"-^ 

32.  ar^  +  2  x^y"  +  2/^"  by  ar^"  -  2  aj"?/"  +  y^. 

An  indicated  product  is  said  to  be  expanded  when  the  multipli- 
cation is  performed. 

Expand : 

33.  (x-\-y)(x  +  y).  39.  (2a^-\-h)(2a^ -b). 

34.  (a-\-h){a-^b).  40.  (a;**  +  3/")  (a;"  —  2/"). 

35.  (c^+d^Xc^  +  d-^).  41.  (3aa;  +  2&y)(3aa;4-2  62^). 

36.  (a;" +  2/")  (a;" +  7/").  42.  (3  oa;  +  2  6y) (3  aa;  -  2  6y). 

37.  (3  a  +  ft) (3 a  +  ft).  43.  (4m  -  5w)(4m  4-5n). 

38.  (3a4-6)(3a— 6).  44.  (« -h  & +  o)(a  +  6  — c). 

45.  (a  +  6)  (a  —  b)(a-\-  h)  (a  -  h). 

46.  (a''  +  x^(a^-a^)(a^  +  af)(a^-a^. 

47.  (a-6)(aH-6)(a2  +  62)(^4_^54)^ 

48.  (a"*  -  6")  (a*^  +  6**)  (a^"  +  6*^). 

49.  (2a  +  36  +  5c)(2a  +  35^5c). 

50.  ('5  m  — 2  71  + a;)  (5  m  — 2  n  — a;). 

51 .  (a;"  —  nx^'-h/  +  i  nx''-^y^)  (x-\-y). 
62 .  (a;^  +  3  a;^-^;?  -  6  a;^-^^^)  {x^-^y^. 


56  ACADEMIC  ALGEBRA 

88.  When  polynomials  are  arranged  according  to  the  ascending 
or  the  descending  powers  of  some  literal  factor,  processes  may 
frequently  be  abridged  by  using  the  Detached  Coefficients. 

63.   Expand  (2a;*-3a^  +  3a;  + l)(3a;  +  2). 

FULL    PROCESS  DETACHED    COEFFICIENTS 

2x*-Sa^-^Sx-{-l  2     -3     +0     +3     +1 

3  a; +2  3+2 


6a^-9x'  +9x'-\-3x  6     -9     +0     +9     +3 

4.x'-6a^  +6a;+2  4     -6     +0     +6    +2 

6  af-5  x*-6  ar'+9  a^+9  x+2  6  x^-5  x'-6  a^-{-9  x^-\-9  x-\-2 

Since  the  second  power  of  x  is  wanting  in  the  first  factor,  the  term,  if 
it  were  supplied,  would  be  0  x^,  and  tlie  detached  coefBcient  of  the  term 
would  be  0. 

The  detached  coeificients  of  missing  terms  should  be  supplied  to  prevent 
confusion  in  placing  the  coefi&cients  in  the  partial  products  and  to  prevent 
errors  in  determining  the  result. 

64.   Multiply  a^ -  2  a^b -{- 3 a^b^  -  5 ab^ -{- 5  b^  by  a  +  2b. 

PROCESS  TEST- 

l_2+3-5+   5  =2 

1  +  2 =3 

1-2+3-5+   5 

2-4  +  6-10  +  10 

1  +  0-1+1-5  +  10  =6 

.   =a'-\-0a'b-aPb^-\-a^b^-5ab*-\-10b' 
=  a'-  a%^  +  a^b^  -5ab'-^  10  b' 

Observe  that  the  powers  of  a  decrease'  uniformly  from  left  to  right,  and 
that  the  powers  of  b  increase  uniformly  from  left  to  right. 

Observe  also  that  the  sum  of  the  exponents  is  the  same  in  every  term. 

Expand  the  following,  using  detached  coefficients,  and  test  the 
results : 

55.    {x^-\-4.a^y-{-6x'y^  +  4:Xy^  +  y^)(x  +  y). 

66.    (5a8-a;7-2ar*  +  a;*  +  3a^-l)(a;  +  2). 


MULTIPLICATION  57 

57.  (a3  4-4a2  +  l6«-32)(as  +  a2  +  a  +  l). 

58.  {p'  -  2pq  +  q")  (f  +  2pq  -\-  (f). 

59.  (a;-l)(a;-2)(a;-3)(a;  +  4)(rc  +  2). 

60.  (15r^s  -  lOrs^  4-  ^  +  s"  +  3.r^s-  +  3r2s3)(r  -  2  s). 

61.  (a;i0-a;9  +  a:8-a;7-t-a;«-a^4-a;^-a^^  +  a^-a;  +  l)(a;-f  1). 

62.  (a;^o  +  3ic8-2a:«4-5a;*  +  2a^  +  2)(x«  +  a;^  +  a^  +  l). 

63.  (ic*  +  a^  +  if^  +  i»  +  1)  (.'c*  -  «^  +  a^  -  a;  +  l)(a;  +  l)(aJ  -  1). 

64.  (a^  -  4aj«y  +  5a;^i/2  ^  SiB^?/*  -  ory  +  a;/  -  y')(2x-3y). 

Expand : 

65.  (2a^4-4ar^  +  8a;  +  16)(3a;-6). 

66.  (a^  +  4ar^  +  5a; -24)(ar^- 4a;  4- 11). 

67.  (a;3_4a^_|_iia._24)(ar'  +  4a;  +  5). 

68.  {2c'-x-\-l){x'  +  x  +  l)(ai'^a^-\-l). 

69.  (16a*-8a3  +  4a2-2a  +  l)(2a  +  l). 

70.  (a3-2a2  +  3a-4)(4a3  4-3a2H-2a  +  l). 

71.  (m*  +  2m3  4-m2-4m-ll)(m2-2m  +  3). 

72.  (m^  +  2  m^n  +  2  mn^  +  w")  (m^  —  2  m^w  +  2  mn'^  —  n^). 

73.  (^-.ix^  +  ix-^\)(x  +  i). 

74.  (Ja;3_^^^_^.|aj  +  2)(a;-J). 

76.  fa'-^Vsa'-\-2aYsa-3\ 

77.  (a;"-^  -  a;"-2  +  a;''-^  -  a;"-'*)  (a;  +  1). 

78.  (!-«'*  +  a^'*  -  a-'''')  (1  -  «")• 

79.  (a^"  -f  2  a'»?>'^  +  ft^")  (««  _  6«). 

80.  (ar^"  —  a;"2/'"  + /'")  (a;"  +  2/"*)- 


58  ACADEMIC   ALGEBRA 

Equations  and  Problems 

89.   1.  Given  5  (2  a; -3)  =7(3  a; +  5)— 72,  to  find  the  value  of  ji. 

Solution 
5(2x-3)=7(3x  +  5)-72. 
Expanding,  10  re  -  15  =  21  x  +  35  -  72. 

Transposing,  10  x  -  21  x  =  15  +  35  -  72. 

Uniting  terms,  —  11  x  =  —  22. 

Multiplying  by  -  1,  11  x  =  22. 

.-.  x  =  2. 
Verification.  —  Substituting  2  for  x  in  the  given  equation, 
5(4 -3)=  7(6 +  5) -72. 
5  =  77  -  72  =  5. 

Find  the  value  of  x,  and  verify  the  result,  in  the  following: 

2.  4  =  5-(a;-2)-(a;-3).         5.   10  a;  -  2(a:  -  3)  =  -  10. 

3.  2  =  2a;-5-(.a;-3).  6.   6a;-3(a;-6)=4(2a:-l)-f2. 

4.  1  =5(2a;-4)-h5.T4-6.  7.   7(5  -  3aj)  =  3(3  -  4x)  -  1. 

8.  4(2-4a;)  =  4-2(a;  +  5). 

9.  5  +  3a;-4  =  13  +  4(a;-4). 

10.  49-2(2a;  +  3)  =  9  +  2(2a?-3). 

11.  3a;-2(4a;-5)=- 2(6  +  2a;). 

12.  3(a;  +  l)-2(2a;  +  5)  =  6(3-a;). 

13.  2(a;-5)  +  7  =  a;H-30-9(x-3). 

14.  5+.7(a;-5)  =  15(a;  +  2-36). 

15.  (a;-2)(x-2)  =  (aj-3)(a;-3)-}-7. 

16.  (a;-4)(a;  +  4)  =  (a;-6)(a;  +  5)  +  25. 

17.  4ar'-4(ar^-.T2  4-a;-2)  =  4a;2. 

18.  7(2a;-36)  =  26-3(2a;  +  &). 

19.  lla=3(a;-2a)-5(2a;-2a). 

20.  3(2  6-4a;)-(a;-6)=-6  6.     24.    3  (a;-a-2  6)=3  ?>. 

21.  4a;-a;2=a;(2-a;)+2a.  25.    5  6=3  (2  a;-6)-4  6. 

22.  2(a;H-d)=10c.  26.    13  (a:-a)=5  (2  a;+a). 

23.  5c=5(a;+a-6).  27.    5  (4a;-3  a)-6  (3  a;-2  a)  =  3  a. 


MUL TI PLICA  TTON  59 

Solve  the  following  problems  and  verify  the  solutions : 

28.  George  and  Henry  together  had  46  cents.     If  George  had 

4  cents  more  than  half  as  many  as  Henry,  how  many  cents  had 

each? 

First  Solution 

Let  re  =  the  number  of.  cents  George  had. 

Then,  a;  —  4  =  the  number  of  cents  George  had  less  4, 

and  2{x  —  4i)=  the  number  of  cents  Henry  had  ; 

.-.  x  +  2(a;-4)=46. 

Expanding,  x  +  2x-8  =  46; 

.-.  X  =  18,  the  number  of  cents  George  had, 

and  2  (x  —  4)  =  28,  the  number  of  cents  Henry  had. 

Second  Solution 
Since  George  had  4  cents  more  than  half  as  many  as  Henry, 
let  2  X  =  the  number  of  cents  Henry  had  ; 

then,  X  +  4  =  the  number  of  cents  George  had, 

and  2x  +  x  +  4=:46; 

.-.  X  =  14, 

2  X  =  28,  the  number  of  cents  Henry  had, 
and  X  +  4  =  18,  the  number  of  cents  George  had. 

Verification 

The  answers  obtained  should  be  tested  by  the  conditions  of  the  problem. 
If  they  satisfy  the  conditions  of  the  problem,  the  solution  is  presumably 
correct. 

1st  condition  :  —  They  had  together  46  cents. 

18  +  28  =  46. 
2d  condition  :  —  George  had  4  cents  more  than  half  as  many  as  Henry. 
18  =  ^  of  28  +  4. 

29.  In  a  certain  election  at  which  8000  votes  were  polled,  B 
received  500  votes  more  than  half  as  many  as  A.  How  many 
votes  did  each  receive  ? 

30.  A  had  $40  more  than  B;  B  had  $  10  more  than  one  third 
as  much  as  A.     How  much  money  had  each  ? 


60  ACADEMIC  ALGEBRA 

31.  Mary  is  25  years  younger  than  her  mother.  If  she  were 
one  year  older,  she  would  be  i  as  old  as  her  mother.  What  is 
the  age  of  each  ? 

32.  If  John  had  3  more  marbles,  he  would  have  3  times  as 
many  as  Clarence.  Both  have  41  marbles.  How  many  has 
each? 

33.  Two  boys  together  had  $  8.20,  and  one  had  50  cents  less 
than  half  as  much  as  the  other.     What  amount  had  each  ? 

34.  If  5  is  subtracted  from  twice  a  certain  number,  and  the 
difference  is  multiplied  by  3,  the  result  is  9  less  than  5  times  the 
number.     What  is  the  number  ? 

35.  A  is  f  as  old  as  B;  8  years  ago  he  was  ^  as  old  as  B. 
What  is  the  age  of  each  ? 

Suggestion.  —  Let  5  x  =  the  number  of  years  in  B's  present  age. 

36.  In  2  years  A  will  be  twice  as  old  as  he  was  2  years  ago. 
How  old  is  he  ? 

37.  Two  wheelmen  start  at  the  same  time  from  A  to  ride  to  B. 
One  rides  at  the  rate  of  10  miles  an  hour,  and  rests  3  hours ;  the 
other  rides  at  the  rate  of  8  miles  an  hour,  and  by  resting  only 
1  hour  arrives  at  B  as  soon  as  the  faster  rider.  How  far  is  it 
from  A  to  B,  and  how  many  hours  are  occupied  in  making  the 
trip  ? 

38.  A  man  had  two  flocks  of  sheep  with  the  same  number  of 
sheep  in  each.  After  selling  100  sheep  from  one  flock,  and  20 
from  the  other,  the  numbers  remaining  were  as  2  to  3.  How 
many  sheep  had  he  in  each  flock  at  first  ? 

39.  Mary  bought  17  apples  for  61  cents.  For  a  certain  num- 
ber of  them  she  paid  5  cents  each,  and  for  the  rest  she  paid  3 
cents  each.     How  many  of  each  kind  did  she  buy  ? 

40.  George  is  J  as  old  as  his  father ;  a  years  ago  he  was  ^  as 
old  as  his  father.     What  is  the  age  of  each  ? 

41.  A  rug  is  3  feet  longer  than  it  is  wide.  When  it  is  placed 
on  the  floor  of  a  certain  room,  it  leaves  a  margin  of  2  feet  on 
every  side.  If  the  area  of  the  floor  is  172  square  feet  greater 
than  that  of  the  rug,  what  are  the  dimensions  of  the  floor  ? 


MUL  riPLICA  TION  61 

SPECIAL  CASES  IN  MULTIPLICATION 

90.  The  square  of  the  sum  of  two  numbers. 

(a  +  6)(a  -t-  h)=  a^  +  2  a6  +  h\ 
{x  4-  y){x  +  2/)=  ^  +  2  a^?/  +  /. 

1.  When  a  number  is  multiplied  by  itself,  what  power  is 
obtained  ?  What  is  the  second  power,  or  square  of  (a  +  6)  ? 
of  (a;  +  2/)? 

2.  How  are  the  terms  of  the  square  of  the  sum  of  two  numbers 
obtained  from  the  numbers  ? 

3.  What  signs  have  the  terms  ? 

91.  Principle.  —  The  square,  of  the.  sum  of  two  numbers  is  equal 
to  the  square  of  the  first  numhevy  plus  twice  the  product  of  the  first 
and  second,  plus  the  square  of  the  second. 

Since  5  a^  x  5  «« =  25  a«,  3  a*6*  x  3  a^¥  =  9  aH^\  etc.,  it  is  evident 
that  in  squaring  a  number  the  exponents  of  literal  factors  are 
doubled. 

Examples 

Expand  by  inspection : 

1.  (m -f  w)(m -f  n).  13.  (3  6 -|- c)^. 

2.  (P  +  9)(i>  +  9)-  14.  (2a  +  3  6)« 

3.  (r  +  s)(r  +  s).  15.  {^x-\-2yf. 

4.  {a  +  x){a^-x).  16.  (Iz  +  ^cf. 

5.  (a5  +  4)(a;  +  4).  17.  (3  6  +  10a;)2. 

6.  (m  +  5)(m  +  5).  18.  (a?-\-hy. 

7.  (a-f-6)(a  +  6).  19.  (««  +  &')'. 

8.  (2/ 4- 7)(2/ +  7).  20.  (a«  +  6«)2. 

9.  (z-fl)(2;  +  l).  21.  (xr+y^f. 

10.  (2  a  +  a;)(2  a  +  a;).  22.    (^a'-\-6hy. 

11.  (3m4-w)(3m  +  n).  23.    (\  +  2a^h)\ 

12.  (5a;-)-2;)(5a;  +  2;).  24.    (a;'^-i  +  2/"-^)'. 


62  ACADEMIC  ALGEBRA 

25.   Find  the  square  of  41. 

Solution 


Square : 

412: 

=  (40  +  1)2  =  402  +  2 

X  40  X  1  +  12  = 

=  1681. 

26.   21. 

29.    45. 

32.    22. 

35.      81. 

27.    24. 

30.    83. 

33.    72. 

36.      91. 

28.    25. 

31.    65. 

34.    43. 

37.    101. 

38.    Find  the 

square  of  71 

Solution 

my 

=  (7  +  ^)2  =  72  +  2  X  7  X 

i+a)2  =  49+7  +  i  =  56i. 

Observe  that  the  middle  term  of  the  square  of  any  number  expressed  by  an 
integer  and  the  fraction  ^  is  equal  to  the  integer.  Hence,  the  square  of  such 
a  number  is  equal  to  the  square  of  the  integer,  +  the  integer,  +  the  square 
of  the  fraction.  Observe  also  that  the  sum  of  the  first  two  terms  of  the  square 
may  be  found  by  multiplying  the  integer  by  the  integer  increased  by  1. 

Thus,  (3|)2  =  9  +  3  +  ^  =  12|, 

or  (3i)2  =  3x4+1  =  12^. 

Find  the  square  of 

39.  51  41.      21  43.    1.5.  45.  6.5. 

40.  41  42.    12^.  44.    5.5.  46.  8.5. 

92.  The  square  of  the  difference  of  two  numbers. 

(a  -  b)(a  -b)=a^-2ab-{-  h\ 
('^  —  y)(^  —  y)=  ^  —  2  xy  -\-  y\ 

1.  What  is  the  square  of  (a  —  b)  ?   of  {x  —  y)? 

2.  How  is  the  square  of  the  difference  of  two  numbers  obtained 
from  the  numbers  ? 

3.  How  does  the  square  of  (a-b)  differ  from  the  square  of 
(a+6)?  ,         • 

93.  Principle.  —  The  square  of  the  difference  of  two  numbers  is 
equal  to  the  square  of  the  first  number,  minus  twice  the  product  of 
the  first  and  second,  plus  the  square  of  the  second. 


MULTIPLICA  TTON 


63 


Examples 

Dxr 

)and  by  inspection : 

1. 

{x  —  m)  (x  —  m). 

10. 

(2a-xy. 

19. 

(3x-2y. 

2. 

(m  —  n)(m  —  n). 

11. 

(Sm-nf. 

20. 

(2x-oyy, 

3. 

(x-6)(x-6y 

12. 

{^x-yy. 

21. 

(2x-^yy. 

4. 

(P-S)(p-S). 

13. 

(om  —  iif. 

22. 

(om-3n)2. 

5. 

(q-7)(q-7). 

14. 

{rtn-4.n)\ 

23. 

{3p-bqy, 

6. 

{a  -  c)  (a  -  c). 

15. 

(p-3qy. 

24. 

(a--h-y. 

7. 

^r-t)(r-ty 

16. 

(a -7  by. 

25. 

(x--y-y. 

8. 

(a  —  x)(a  —  x). 

17. 

(4  a -3)1 

26. 

^^m-l_yn-iy^ 

9. 

(x-l)(x-l). 

18. 

(5  a; -4)2. 

27. 

(maj"*  —  ny'^y. 

28.  Find  the  square  of  19. 

Solution 

192  =  (20  -  1)2  =  202  -  2  X  20  X  1  +  12  =  36I. 

Find  the  square  of 

29.  49.  32.  29.  35.  67. 

30.  69.  33.  38.  36.  89. 

31.  79.  34.  48.  37.  99. 


38.  998. 

39.  999. 

40.  595. 


94.   The  square  of  any  polynomial. 

By  actual  multiplication, 

(a-^b-\-cy=a^-{-b^-{-c'-h2ab-{-2ac+2bc. 

(^a+b-c+dy=a^-{-b^-\-(^-\-(jP-\-2  ab-2ac-\-2  ad-2  bc-^2  bd-2  cd. 

(a+.-.+A;H f- m)-  =  a- +  •••  +  ^^^H \-m^ -\-2ak-\- "• 

-\-2a7n-\ 1-2 /cm  +  •••. 

1.  In  the  square  of  each  polynomial  what  terms  are  squares  ? 

2.  How  are  the  other  terms  formed  from  the  terms  of  the 
polynomial  ? 

3.  What  signs  have  the  squares  ?     How  are  the  signs  of  the 
other  terms  determined  ? 


64  ACADEMIC  ALGEBRA 

95.  Principle.  —  The  square  of  *a  polynomial  is  equal  to  the 
sum  of  the  squares  of  the  tenns  and  timce  the  product  of  each  term 
by  each  term  that  follows  it 

Examples 
Expand  by  inspection : 

1.  {x  +  y-\-zy.  4.    {x-y-\-zf.  7.    (a  -  2  6  +  c)^. 

2.  {x-{-y-z)\  5.    (x  +  y-^zf.  8.    {2a-h-c)\ 

3.  (x-y-zf.  6.    (x-y  +  ^zf.  9.    (6-2a  +  c)l 

10.  {ax  —  hy-\-czY.  15.  (2a  —  5 />  +  3c)2. 

11.  (ma  +  nb  -  rzf.  16.  (^2m  —  4:n-  rf. 

12.  (qb-pc-rdy.  17.  a2-2a^4-32/)l 

13.  (ac  —  bd-def.  18.  (ft  +  m  +  6  +  n)2. 

14.  (3a;-2  2/  +  4;s)2.  19.  (a  -  7n -^  b  -  nf. 

96.  The  product  of  the  sum  and  diiference  of  two  numbers. 

(a  -^b)(a-b)  =  a^-  b\ 
(of  +  2/")  C^^**  —  2/")  =  a^"  —  2/^". 

1.  Since  a  +  6  represents  the  sum  and  a  —  b  the  difference  of 
two  numbers,  to  what  is  the  product  of  the  sum  and  the  difference 
of  two  numbers  equal  ? 

2.  How  are  the  terras  of  the  product  obtained  from  the 
numbers  ? 

3.  What  sign  connects  the  squares? 

97.  Principle.  —  The  product  of  the  sum  and  difference  of  two 
numbers  is  equal  to  the  difference  of  their  squares. 

Examples 
Expand  by  inspection : 

1.  (x-\-y)(x-y).  6.  (r -]- s)(r  -  s), 

2.  (a  +  c)(a—c).  7.  (a;  +  1) (a;  —  1). 

3.  (m -{- n)  (m  —  n).  8.  (a^  +  1)  (ar  —  1). 

4.  (p  +  q)(p  -q)-  9.  (a^+  1) (x'  -  1). 

5.  (i>  4- 5)  (i>  -  5).  10.  (a;^  -  1)  (a;^  H-l). 


MUL  TIPL ICA  TION  65 

11.  (ar^-l)(ar'+-l).  20.  (2  a^ -\- 5  f)  (2  a^  -  5  f). 

12.  (x^  +  f)(x^-f).  21.  (3a^-{-2f)(3a^-2f). 

13.  (a6  +  5)(a6-5).  22.  (2 a" -^  2  b')  (2 a' -  2  b^), 

14.  (cd  +  (Z2>)(cd_d2).  23.  (-5n-2b)(-5n  +  2b). 

15.  (2a;  +  32/)(2a;-32/).  24.  (- a;  -  2y)(- a;  +  2y). 

16.  (3m  +  4n)(3m-4n).  25.  (- 5  -  3m)(- 5  +  3m). 

17.  (12  +  xy)(12  —  xy).  26.  (iC^-^  +  2/"+^  (a:"*  "^  —  2/**+^). 

18.  (3m2n-6)(3m2n  +  6).  27.  (3af*  +  72/")(3a;'"  -  Ti/"). 

19.  (a6  +  c(«)  (a6  -  cd).  28.  (5  a'b^  +  2  of)  (5  a'b' -  2  of). 
One  or  both  of  the  numbers  may  consist  of  more  than  one  term. 

29.  Expand  (a  -^  m  —  n)  (a  —  m  -{-  n). 

Solution 

a  -\-  m  —  n  =  a  -\-  (m  —  n). 

a  —  m  +  n  =  a  —  (m  —  n). 

.'.  [a -\- m  -  n'}la  -  m  +  n]  =  [a  +  (m  —  n)'\la  - (wi  -  w)]. 

Prin.,  =  a2  _  (,^  _  n)2 

§93,  =a2_(,„2_2nin  +  n2) 

=  a2  -  w2  +  2  mn  -  n2. 
Expand : 

30.  (a-\-x-y){a^x-\-y).  36.  (y H-c +  d)(2/ +  c-cT). 

31.  (x-^c  —  d)(x  —  c-{-d).  37.  (a -f  a;  +  2/)  (a  4- a?  —  y). 

32.  (r4-l>-9)(r-i9  +  g).  38.  (a^ -\-2x +  l)(a^ +  2  x -1). 
33.'  (r-hp  +  q)(r-p-q).  39.  (a^  +  2 a;  —  1) (ar^ -  2 a;  -f-  !)• 

34.  (x-^b  +  n)(x-b-n).       40.    (a;^  +  3 a;  -  2) (ar^  -  3  a^  +  2). 

35.  {y-{-c  +  d)(y-c-d).        41.    (a;^  +  3  a;  +  2)  (ar^  -  3  a;  +  2). 

42.  (m*-2m2  +  l)(m*  +  2m2  +  l). 

43.  (2a;H-32/-42)(2a;  +  32/  +  4z). 

44.  (2a^-a;y  +  32/0(2a^H-a;2/-32/2). 

45.  (x'  +  xy-^y^){x'-xy-{-y'). 

▲CAD.    ALG. — 6 


66  ACADEMIC  ALGEBRA 

46.  [(a4-&)4-(c  +  (^)][(a  +  6)-(c  +  c?)]. 

47.  (a -j- b -{- X -\- y)(a -\- b  —  X  —  y). 

48.  (a  -f  5  +  m  —  w)  (a  +  6  —  m  -f-  ti). 

49.  (ic  —  m  +  2/  —  n)  (a;  —  w  —  2/  4-  w). 

50.  (p  —  q  -h  r  -^  s)  (p  —  q  —  r  —  s). 

51 .  (a  —  m  —  6  —  ?i)  (a  -f-  m  —  6  +  n). 

52.  (a-{-x-\-b  —  y)(a  —  x-\-b  +  y). 

53.  Find  the  product  of  32  x  28. 

Solution 
32  X  28  =  (30  +  2)  (.30  -  2) 

=  302  -  22  =  900  -  4  =  896. 

Find  the  product  of 

54.  31  X  29.  57.  48  x  52.  60.  98  x  102. 

55.  42x38.  58.  57x63.  61.  99x101. 

56.  69  X  71.  59.  95  x  85.  62.  95  x  105. 

63.  What  is  the  square  of  95  ? 

Solution 
(a  +  &)  (a  -  6)  =  a2  _  ^,2.  (l>j 

Transposing,  etc.,     .  a^  =  («  +  &)(«-&)  +  b^.  (2) 

Let  a  =  95  and  b  =  5. 

Equation  (2)  becomes         952  =  (95  +  5)  (95  -  5)  +  52 

=  100  X  90  +  25  =  9025. 

64.  What  is  the  square  of  48  ? 

Solution  , 

Let  a  =  48  and  6  =  2. 

Equation  (2)  becomes         482  =  (48  +  2)  (48  -  2)  +  22 

=  50  X  46  +  4  =  2304. 

Square  by  a  similar  process : 

65.  98.  67.    93.  69.    58.  71.    87.  73.      68. 

66.  96.  68.    97.  70.    49.  72.    79.  74.    129. 


MUL  TIPLICA  TION  67 

98.  The  product  of  two  binomials  that  have  a  common  term. 

(x  +  2)(x  -  5)=  a;2  +  2  a;  -  5  a;  -  10 
==  a^  _  3  a;  _  10. 

(a;  -  2)(x  -5)=x'-2x-5x-\-10 

=  a^  _  7  a;  +  10. 

(x  -\-  a)(x  —  b)  =  ay^  -\-  ax  —  bx  —  ab 
=  x^  -\-(a  —  b)x  —  ab. 

1.  How  many  terms  are  alike  in  each  pair  of  factors  ? 

2.  How  is  the  first  term  of  each  product  obtained  from  the 
binomial  factors  ? 

3.  How  is  the  third  term  of  each  product  obtained  from  the 
factors  ? 

4.  How  is  the  second  term  of  the  product  in  the  first  example 
obtained  from  the  factors  ?  in  the  second  example  ?  in  the  third 
example  ?  in  the  fourth  example  ? 

5.  How  can  the  formation  of  the  second,  or  middle  term  be 
described  so  as  to  apply  to  all  of  the  examples  ? 

99.  Principle.  —  The  product  of  tioo  binomials  thai  have  a  com- 
mo7i  term  is  the  algebraic  sum  of  the  square  of  the  common  term,  the 
common  term  multiplied  by  the  algebraic  sum  of  the  unlike  terms, 
and  the  algebraic  product  of  the  unlike  terms. 

Examples 

Expand  by  inspection : 

1.  (a;4-5)(a;  +  6).  7.  {x-b)(x-l), 

2.  (a;  +  7)(a; -f  8).  8.  (a;  +  5)(a;  +  8). 

3.  (.T  -  7)(a;  +  8).  9.  (j9  -  4)(i)  4- 1). 

4.  (a;  +  7)(a;  -  8).  10.  (r-3)(r-l). 

5.  (a;  -  5)(a;  -  4).  11.  (m  -  6)(m  +  5). 

6.  {x  -  3)(a;  -  2).  12.  (m  -  2)(m  +  10). 


68  ACADEMIC  ALGEBRA 

13.  (n-S)(n-12).  ■  24.  (j^ -2a)(y  +  3b), 

14.  (n- 6)  (71 +  15).  25.  (z -4  a)  (2  + 3  a). 

15.  (a^  +  5)(x'-S).  26.  (2 a;  +  5) (2 a;  +  3). 

16.  (a^-7)(a^  +  6).  27.  (2 x - 7) (2 a;  +  5). 

17.  (aj^-3)(aj^  +  9).  28.  (3 1/ - 1) (3 2/ +  2). 

18.  (x-{-c)(x  +  d).  ^29.  (4a^  +  l)(4a^-7). 

19.  (m-^d)(m-\-b).  30.  (a6 - 6) (a6  +  4). 

20.  (r4-a)(^-&).  31.  (a^y  -  a)  (a^V  +  2  a). 

21.  (s  —  a)  (s -{- n).  32.  (2  m  —  a&)  (2  7«,  +  3  a&). 

22.  (a;" -5)  (a;" +  4).  33.  (5p- ac^)(2a€^ -\-5p). 

23.  (a;"-a)(aj»-6).  34.  (Sxy-\-y^)(y^-xy). 


35.    (a4-6  +  5)(a  +  6  +  2). 


36.    (a-6-4)(a-6  +  10). 


37.    (a;  +  2/-l)(a^  +  2/  +  2). 


38.    (aj_2/_2)(a;-2/-8). 


39.    (a^-f-a;-l)(a;2  +  a;4-3). 


40.    (2m  +  n-3)(2m  +  7i  +  4). 

By  an  extension  of  the  method  given  above,  the  product  of 
any  two  binomials  may  be  written. 

41.  Expand  (3x-\-2y)(ox- 4:y). 

Solution 

(3  2c  +  2y)(5x-4y)=  15a;2  +  lOxy  -  12xy  -Sy^ 
=  15  x2  -  2  icy  -  8  y'^. 

Expand  by  inspection : 

42.  (2x-\-5y)(3x-\-4.y),  45.    (3  a^  _  1)  (2  a^  +  3). 

43.  (Sx-4.y)(2x-^5y),  46.    (m^ -^  n)  (2  m' -  71). 
"44.    (3a-66)(2a4-36).  47.    (a  +  2  6)(c  -  2d).. 


MUL  TIPLICA  TION  69 


Exercises 

100.  1.  In  a  certain  family  there  are  three  children  each  of 
whom  is  2  years  older  than  the  one  next  younger.  When  the 
youngest  is  x  years  old,  what  are  the  ages  of  the  others  ?  When 
the  oldest  is  y  years  old,  what  are  the  ages  of  the  others  ? 

2.  What  two  whole  numbers  are  nearest  to  the  whole  num- 
ber X? 

3.  Mary  read  10  pages  in  a  book,  stopping  at  the  top  of  page  a. 
On  what  page  did  she  begin  to  read  ? 

4.  A  man  made  three  purchases  of  a,  h,  and  2  dollars,  respec- 
tively, and  tendered  a  10-dollar  bill.  Express  the  number  of  dol- 
lars change  due  him. 

5.  A  sold  B  grain,  hay,  and  potatoes  for  a,  b,  and  c  dollars, 
respectively ;  but  some  of  the  grain  becoming  damaged,  and 
some  of  the  potatoes  having  been  frozen,  he  deducted  x  +  y 
dollars  from  B's  indebtedness.  If  B  offered  a  lOO-doUar  bill  in 
payment,  what  was  the  amount  due  him  in  return  ? 

6.  What  is  the  cost  of  5  apples  at  b  cents  each  ?  What  will 
a  apples  cost  at  b  cents  each  ? 

7.  How  many  cents  are  there  in  a  dollars?  How  many 
dimes  are  there  in  b  dollars  ?   in  oo;  dollars  ? 

8.  If  a  man  earns  $2  per  day,  how  much  will  he  earn  in 
a  days  ?   in  c  days  ?   in  a  —  c  days  ? 

9.  How  much  will  a  man  whose  wages  are  a  dollars  per  day 
earn  in  b  days  ?   in  c  days  ?   in  a;  days  ?   in  a  days  ? 

10.  If  a  man  earns  a  dollars  per  month  and  his  expenses  are 
b  dollars  per  month,  how  much  will  he  save  in  a  year  ? 

11.  How  far  can  a  wheelman  ride  in  a  hours  at  the  rate  of 
b  miles  an  hour  ?  How  far  will  he  have  ridden  after  a  hours,  if 
he  stops  c  hours  of  the  time  to  rest  ? 

12.  A  has  X  dollars  and  B  y  dollars.  If  A  gives  B  m  dollars, 
how  much  will  each  then  have  ? 

13.  The  number  25  may  be  written  20  -f  5.  Write  the  number 
whose  first  digit  is  x  and  second  y. 


70  ACADEMIC  ALGEBRA 

14.  A  book  contained  x  pages.  If  they  averaged  y  lines  to  a 
page  and  z  letters  to  a  line,  how  many  letters  were  there  in  the 
book? 

15.  How  many  square  rods  are  there  in  a  square  field  one  of 
whose  sides  is  2  6  rods  long  ?   x  rods  long  ? 

16.  What  is  the  number  of  square  rods  in  a  rectangular  field 
whose  length  is  30  rods  and  width  20  rods  ?  What  will  be  the 
area  in  square  rods,  if  the  length  is  a  rods  and  the  width  b  rods  ? 

17.  A  fence  is  built  across  a  rectangular  field  so  as  to  make 
the  part  on  one  side  of  the  fence  a  square.  If  the  field  is  a  rods 
long  and  b  rods  wide,  what  is  the  area  of  each  part  ? 

18.  A  man  who  had  s  dollars  bought  b  bales  of  hay  at  n  cents 
a  bale  and  a  bushels  of  oats  at  m  cents  a  bushel.  How  many 
cents  had  he  left  ? 

19.  A  speculator  bought  a  car  loads  of  wheat  at  m  dollars  a 
car,  and  sold  b  car  loads  of  it  at  n  dollars  a  car.  How  much  did 
he  gain  by  the  transaction,  if  he  sold  the  rest  of  the  wheat  for 
c  dollars  a  car  ? 

20.  A  sold  a  farm  which  had  cost  him  ii  dollars  an  acre  to  two 
men,  a  acres  to  one  and  b  acres  to  the  other,  at  the  uniform  price 
of  m  dollars  an  acre.     How  much  did  he  gain  ? 

"21.  In  a  library  there  were  p -\-  q  volumes  that  averaged  p -\- q 
pages  per  volume,  p  -\-  q  words  per  page,  and  7  letters  per  word. 
How  many  letters  were  there  in  all  these  books  ? 

22.  A  wheelman  who  had  a  journey  of  x  miles  to  make  rode 
the  first  a  hours  at  the  rate  of  b  miles  an  hour,  when  he  was 
obliged  to  stop  c  hours  for  repairs.  After  that  he  rode  2  miles 
an  hour  faster,  so  that  he  made  the  whole  journey  in  10  hours. 
What  was  the  length  of  the  journey  ? 

23.  A  wheelman  rode  a  hours  at  the  rate  of  m  miles  an  hour, 
then  decreased  his  speed  5  miles  an  hour  for  3  hours,  and  finished 
his  journey  in  b  hours  more,  increasing  his  first  rate  2  miles  an 
hour.     How  far  did  he  ride  ? 

If  a  =  4,  6  =  2,  and  m  =  10,  how  many  miles  did  he  ride,  and 
in  what  time  did  he  accomplish  the  journey  ? 


DIVISION 


101.    1.   Since  +  2  times  +  10  is  +  20,  if  +  20  is  divided  by 
4- 10,  what  is  the  sign  of  the  quotient  ? 

2.  What  is  the  sign  of  the  quotient  when  a  positive  number 
is  divided  by  a  positive  number  ? 

3.  Since  +  2  times  —  10  is  —  20,  if  —  20  is  divided  by  —  10, 
what  is  the  sign  of  the  quotient  ?  if  —  40  is  divided  by  —  5  ? 

4.  What  is  the  sign  of  the  quotient  when  a' negative  number  is 
divided  by  a  negative  number  ? 

6.    What  is  the  sign  of  the  quotient  when  the  dividend  and 
divisor  have  like  signs  ? 

6.  Since  +  2  times  -  10  is  —  20,  if  -  20  is  divided  by  -f-  2. 
what  is  the  sign  of  the  quotient  ?  if  —  20  is  divided  by  -|-  5  ? 

7.  Since  -  2  times  -  10  is  +  20,  if  -f  20  is  divided  by  -  2, 
what  is  the  sign  of  the  quotient  ?  if  -}-  20  is  divided  by  —  4  ? 

8.  What  is  the  sign  of  the  quotient  when  the  dividend'  and 
divisor  have  unlike  signs  ? 

9.  How  many  times  is  2  a  contained  in  6  a  ?  in  10  a  ? 
How  is  the  coefficient  of  the  quotient  obtained  ? 

10.  Since  a^  x  a*  =  a^,  if  a*  is  divided  by  a^  what  is  the  quo- 
tient ?     What  is  the  quotient,  if  a®  is  divided  by  a^  ? 

What  is  the  quotient,  if  6^  is  divided  by  b^  ?  by  b*? 

How  is  the  exponent  of  a  number  in  the  quotient  obtained  ? 

11.  What  is  the  exponent  of  a  in  the  quotient  of  a*  -i-  a*?  of 
a^  -t.  a^9     How  many  times  is  a*  contained  in  a'^  ?  o?  in  a?  ? 

12.  What  is  the  value  of  any  expression  whose  exponent  is  0? 

71 


72  ACADEMIC  ALGEBRA 

102.  In  multiplication  two  numbers  are  given  and  their  product 
is  to  be  found.  The  inverse  process,  finding  one  of  two  numbers 
when  their  product  and  the  other  number  are  given,  is  called 

Division. 

10  -f-  2  =    5  and  D ---  d  =  q 

are  inverses  of  5  x  2  =  10  and  q  x  d  =  D. 

The  dividend  corresponds  to  the  product,  the  divisor  to  the  mul- 
tiplier, and  the  quotient  to  the  multiplicand.  Hence,  the  quotient 
may  be  defined  as  that  number  ivhich  multiplied  by  the  divisor 
produces  the  dividend. 

The  quotient  of   a  divided   by  b,  indicated   by  (a  h-  b),  or  -, 

is  defined  for  all  values  of  a  and  b  by  the  relation 

?  X  &  =  a. 
b 

103.  Principles.  —  1.  Law  of  Signs.  —  Tlie  sign  of  the  quotient 
is  -f-  when  the  dividend  and  divisor  have  like  signs,  and  —  when 
they  have  unlike  signs. 

2.  Law  of  Coefficients.  —  The  coefficient  of  the  quotient  is  equal  to 
the  coefficient  of  the  dividend  divided  by  that  of  the  divisor. 

3.  Law  of  Exponents.  —  The  exponent  of  a  letter  in  the  quotient 
is  equal  to  its  exponent  in  the  dividend  diminished  by  its  exponent  in 
the  divisor. 

An  expression  whose  exponent  is  0  is  equal  to  1, 
The  Law  of  Signs  may  be  established  as  follows : 
'  Since  +  a  x  +  b  =  -\-  ab,  +  ah  -^ -\- b  = -{■  a. 

Since  -}-  a  x  —  b  =  -  ab,  —  ab  -r-  —  b  =+  a. 

Since  -  ax  +  b  =—  ab,  -  ab  i- -{-  b  =—  a. 

Since  -  a  x-b=+  ab,  +  ab  -^-b=-  a. 

The  Law  of  Exponents  or  the  Index  Law  for  Division  may 
be  established  as  follows,  m  and  7^  being  positive  integers  and  m 
being  greater  than  n : 

By  §  24,        a""  =  a  X  a  X  a---to  m  factors, 
a^  =  a  X  a  X  a  '"  to  n  factors  ; 
.-.  a*"  -^-  a»  =  (a  X  a  X  a  •••  to  m  factors)  -^(a  x  a  x  a '--to  n  factors) 
=  a  X  a  X  a  •"  to  {m  —  n)  factors. 
Hence,  a"^ -^  a"^  =  a"»-". 


DIVISION  73 

104.  Commutative,  Associative,  and  Distributive  Laws  for  Division. 

1.  The  Commutative  Law  may  be  established  as  follows : 
Since,  by  the  definition  of  division,  §  102,  a  =  a  -^  c  x  c, 

axb-i-c  =  a-^cxcxb-i-c 
§82,  =a^cxbxc-^c 

=  a-^  c  X  b.  (1) 

Also,  §  102,  a-i-b-^c  =  a-^cxc^b^c 

by  notation,  §  29,  =  ^a -^  c  x  c  -4-  6)  h-  c 

by  (1),  =(aT^-^  6  X  c)h-c 

by  notation,  '  =  a  ^  c -i-b  x  c -^  c 

=  a^c^b.  (2) 

The  Commutative  Law  for  division  is  expressed  by  (2). 

(2)  may  be  written  ^  h-  c  =  -  -^  6.      (1)  may  be  written  —  =  -  x  6. 
be  c      c 

It  follows  from  (1)  and  (2)  that  in  a  succession  of  multiplications  and 

divisions  the  multipliers  and  divisors  may  be  arranged  in  any  order. 

2.  The  Associative  Law  may  be  established  as  follows: 
By  the  Commutative  Law  just  proved, 

a  X  b  -i-  c  =  b  -^  c  X  a 

by  notation,  §  29,  =  (6  -h  c)  x  a 

by  the  Commutative  Law,  =  a  x  (6  -4-  c).  (3) 

Also  a-T-6->c  =  6xc-^(6xc)xa-f-6-f-c 

by  the  Commutative  Law,  =b-^bxc^cxa^{bxc) 

=  a^(6xc).  (4) 

The  Associative  Law  for  division  is  expressed  by  (4). 

(4)  may  be  written  ^  -  c  =  —  •      (3)  may  be  written  ^  =  a  x  -• 
b  be  c  c 

It  follows  from  (3)  and  (4)  that  in  a  succession  of  multiplications  and 

^divisions  the  multipliers  jxnd  divisors  may  be  grouped  in  any  manner,  each 

element  keeping  its  own  sign,  x  or  -^,  if  the  sign  x  precedes  the  sign  of 

grouping,  but  changing  it,  if  the  sign  -f-  precedes  the  sign  of  grouping. 

3.  The  Distributive  Law  may  be  established  as  follows : 

§85,  (a^m  +  b^m)xm  =  a^mxm-\-b-^mxm 

=  a-\-b. 
Dividing  both  members  by  w. 


a-i-m  +  b-^m=(a  +  b')-^m; 

.  V,  X  •  a  +  b      a  ,    b 

that  IS,  — ^ —  =  — I 

m        mm 


74  DIVISION 

105.   The  Reciprocal  of  a  number  is  1  divided  by  the  number. 
The  reciprocal  of  5  is  - ;  of  6,  -  ;  of  (a  +  6), 


5'  h  ^  '    a  +  b 

106.  'aH-&==axlH-6 
by  the  Associative  Law,  =  a  x  (1  h-  &). 

Hence,  dividing  by  a  number  is  equivalent  to  multiplying  by  the 
reciprocal  of  the  number. 

107.  To  divide  a  monomial  by  a  monomial. 

Examples 

1.   Divide  -ISa'b^  by  6ab\ 

Explanation.  —  Since  the  dividend  and  divisor  have 
PROCESS  unlike  signs,  the  sign  of  the  quotient  is  —  (Prhi.  1). 

6ab-}—lSa^  Then,  -  18  divided  by  6  is  -  3  (Prin.  2);  a^  divided 

—  3a^6        ^y  ^  ^^  ^*  (Prin.  3);  and  b^  divided  by  b^  is  b  (Prin.  3). 
Therefore,  the  quotient  is  —  3  a*b. 

Rule.  —  Divide  the  numerical  coefficient  of  the  dividend  by  the 
numerical  coefficient  of  the  divisor,  and  to  the  result  annex  the  letters, 
each  with  an  exponent  equal  to  its  exponent  in  the  dividend  mirius 
its  exponent  in  the  divisor. 

Write  the  sign  +  before  the  quotient  when  the  dividend  and  divisor 
have  like  signs,  and  the  sign  —  when  they  have  unlike  signs. 


2. 

3. 

4. 

5. 

6. 

Divide 

)      12a; 

-12arV 

3b  xf 

-14a^/ 

-26a2W 

By 

4.x 

4  0^/ 

-Ixy 

-2o?y 

ISabh 

Find  the  quotient  of 

7.  28  a*62c -5- -  7  a6c.  10.  -Ua^fz^  ^1  xyz\ 

8.  -l^a^f^^^xyz.  11.  -  27 mVj9« -^ 9 m^n^. 

9.  -36a%V---9mV.  12.  -  3^  (fr'p  ^  - 13  qp. 

108.    To  divide  a  polynomial  by  a  monomial. 

Examples 
1.    Divide  5  iry  -  10  o.^  +  5  ar^/  by  5a^2/3 

PROCESS.  §  104,  3,     5  x'f)^  xY  -  10  a^y"  -h  5  x^lf 


DIVISION  75 

Rule.  - — Divide  each  term  of  the  dividend  by  the  divisor,  and  find 
the  algebraic  sum  of  the  partial  quotients. 

Find  the  quotient  of 

4  a^b  4  mn 

24a^5^  +  32a%'^-4Da^5^  5  a;^y  -  10  arV  +  20  a.^ 

8a^62  *  •  bxy 


8. 


-  35  x'f;^  +  45  a^V^^ 
5a;^2/^2 

-39a;^yV  +  65a:^y^z^ 
-13ary2« 

25r«s«-125r^g«-75  7-V'^ 
5?V 

27c»d^-39c*d«-42c»d* 


—a—b—c—d—e 

-1 

—  a  —  a-b  —  ah  —  aH  - 

-a'e 

10. 


12.  (34  aV2/2- 51  aVy*- 68  aVy«) -J- 17  a-ar^/. 

13.  (8  a'b^  -  28  a«6*  -  16  a'b'  +  4  a*6«)  -«-  4  a*6». 

•       14.  [a{b-cf-bib-cy  +  c{b-c)']-^{b-c). 

15.  [(x  -  2/)  -  3 (a;  -  ^/)2  +  4a;(a;  -  yf]  -  (a;-  y). 

16.  (af  -  2  «•+»  -  5  «•+*  -  af +'^  +  3  a-+^)  --  a;«. 

17 .  (2/"+^  —  2  y^-^^  —  tp^^  —  3  y"+*  +  ?/'•+')  -H  ?/"+'. 

18.  (a;"  —  a;"-^  +  a;"-^  —  a;'-^  +  a;""*  —  af"^)  -^  ar^. 

109.    To  divide  a  polynomial  by  a  polynomial. 

Examples 
1.   Divide  3ar^  + 35  + 22a;  by  a? +  5. 


PROCESS 

3a^+22a;  +  35 

7  a; +  35 
7a; +  35 

x  +  B 

TEST 

+  60  --  +  6 

3a;(a;  +  5)     . 

3a;  +  7 

=  +  10 

7(a;  +  5)     . 

76  ACADEMIC  ALGEBRA 

Explanation.  — For  convenience,  the  divisor  is  written  at  the  right  of 
the  dividend,  and  both  are  arranged  according  to  the  descending  powers  of  x. 

Since  the  dividend  is  the  product  of  the  quotient  and  divisor,  it  is  the 
algebraic  sum  of  all  the  products  formed  by  multiplying  each  term  of  the 
quotient  by  each  term  of  the  divisor.  Therefore,  the  term  of  highest  degree 
in  the  dividend  is  the  product  of  the  terms  of  highest  degree  in  the  quotient 
and  divisor.  Hence,  if  Sx^,  the  first  term  of  the  dividend  as  arranged, 
is  divided  by  x,  the  first  term  of  the  divisor,  the  result,  3  x,  is  the  term  of 
highest  degree,  or  the  first  term,  of  the  quotient. 

Subtracting  Sx  multiplied  by  (aj  +  5),  or  3x  times  (x  +  5)  from  the  divi- 
dend, the  remainder  is  7  x  +  35. 

Since  the  dividend  is  the  algebraic  sum  of  the  products  of  each  term  of  the 
quotient  multiplied  by  the  divisor  and  since  the  product  of  the  first  term  of  the 
quotient  multiplied  by  the  divisor  has  been  canceled  from  the  dividend, 
the  remainder,  or  new  dividend^  is  the  product  of  the  other  part  of  the'quotient, 
multiplied  by  the  divisor. 

Proceeding,  then,  as  before,  7  x  -4-  x  =  7,  the  next  term  of  the  quotient. 
7  multiplied  by  (x  +  5) ,  or  (x  +  5)  multiplied  by  7  equals  7  x  +  .35.  Sub- 
tracting, there  is  no  remainder.  Hence,  all  of  the  terms  of  the  quotient  have 
been  obtained,  and  the  quotient  is  3  x  -f  7. 

Test. — Let  x  =  1. 

Dividend  =  3  x2  +  22  x  +  35  =  3  +  22  +  35  =  +  60. 

Divisor  =x  +  5  =1  +  5  =+6. 

Quotient  should  be  equal  to  +  10. 

Quotient  =3x  +  7  =3  +  7  =+10. 

Similarly,  the  result  may  be  tested  by  substituting  any  other  value  for  x. 
When  the  value  substituted  for  x  gives  the  result  0  -h  0  or  0  for  a  divisor, 
some  other  value  should  be  tried. 

Rule.  —  Arrange  both  dividend  and  divisor  according  to  the 
ascending  or  the  descending  poivers  of  a  common  letter. 

Divide  the  first  term  of  the  dividend  by  the  first  term  of  the  divisor, 
and  write  the  result  for  the  first  term  of  the  quotient. 

Multiply  the  whole  divisor  by  this  term  of  the  quotient,  arid  sub- 
tract the  product  from  the  dividend.  Tlie  remainder  will  be  a  neiv 
dividend. 

Divide  the  new  dividend  as  before,  and  continue  to  divide  in  this 
way  until  the  first  term  of  the  divisor  is  not  contained  in  the  first 
term  of  the  new  dividend. 

If  there  is  a  remainder  after  the  last  division,  write  it  over  the 
divisor  in  the  form  of  a  fraction,  and  add  the  fraction  to  the  part 
of  the  quotient  previously  obtained. 


DIVISION 


11 


2.   Divide  2  a*  -  5  af^b  -^  6  a^b^  -  4:  ab^  -{- b*  by  a^-ab-\-b\ 


PROCESS                                                    TEST 

2a*  -5a^b  -\-6a^b'  -4.ab^+b* 

a^-    ab-\-b^        0  -- 1 

2a*-2a;'b-h2a'b' 

2a^-3ab-\-b'          =0 

-Sa^b-^-Sa'b^-Sab^ 

a^b^-    ab^  +  b* 

3.   Divide  a* -\- 9  a^ -\- SI  by  a^-Sa-^-d. 

PROCESS                                       TEST 

a4_f_9^2^81                       1 

a«-3aH-9         91-?- 7 

a<_3a3  +  9a*                     1 

a2  +  3a4-9          =13 

3a3  +  81 

Sa^-9a^  +  27a 

9a2-27a4-81 
9a2-27a  +  81 

Divide,  and  test  the  results  : 

4.  a^-x-20  by  a?  — 6. 

6.  or'  4-  7  a;  +  12  by  .T  -h  3. 

6.  m^  -  3  m  -  18  by  m  -  6. 

7.  i*-ei2^w  by  l^-h2. 

8.  cc?  -  c?2  4-  2  c^  by  c  -h  (i. 

9.  ic2-lla;  +  10  by  a;  -  10. 

10.  0)2 +  15  a;  4- 54  by  x-i-6. 

11.  v-^n  _^  11  ^«  +  30  by  r^-\-6. 

12.  aW  —  4  am^  +  3  m^  by  am  —  1. 

13.  6a2  +  13a6  +  662  by  3a  +  26. 

14.  a*  +  16  +  4  a2  by  2  a  +  a^  +  4. 

15.  a^+a^  +  a*  +  a^  +  3a-l  hy  a  +  1 

16.  20a:2y-25ar'-182/3-f  27a.y  by  62/ -5x. 


T8  ACADEMIC  ALGEBRA 

17.  aa^  —  aV  —  boc^+b^  by  ax—  b, 

18.  a*-41a-120  by  a2  +  4a-h5. 

19.  ar^-61a;-60  by  x^-2x-S. 

20.  25a;5-aj3-8a;-2x2  by  5a^-4aj. 

21.  41/^ -9?/'  + 62/ -1  by  2y^  +  3y-l. 

22.  a*-4a3a;  +  6aV-4aa^  +  a;^  by  a2-2aa;  +  aj*. 


23.    a^-1 
a^  +  a* 


24 


a*  —  a^  _^  a^  —  a  +  1  H- 


-2 
a  +  l 


g^  +  g^ 

a  +  1 
-2 


a^  +    a?  -  25 
a^-3ar^ 


3a^+    a: 
Sx'-dx 


x-S 

a^  +  3a;4-10  4-— -„ 
X  —  o 


10a;-25 
10a; -30 


25.    a;*-3a,^+    a;2  4-2a; 
X*-    a?-2x^ 


-2a?-{-3x'  +  2x 
-2x^^2x'  +  4:x 

x'-2x-\ 
a^-    a?-2 


x^-    x-2 


x^-x-2 


-    «+l 


DIVISION 


79 


Divide : 

26.  m'^  +  n*  by  m  +  n. 

27.  ar^  + 32  by  a; +  2. 

28.  Q^  +  y'^  by  x^  +  ]f. 

29.  a^  4- 5  tt"  —  a^  +  2  a  +  3  by  a  —  1. 

30.  2w^-4n*-3n»H-7n2-3n-|-2  by  n-2. 

31.  a^ -\-  }/  +  z^  —  3  xyz  by  x-\-y  -\-z. 

32.  m'  +  71^  +  a^  +  3  m^;i  +  3  m?i^  by  m  4-  n  -I-  a. 

33.  a3_6a2_^12a-8-6=*  by  a-2-6. 

34.  f-{-3y'  +  ^f  +  3y-  +  3y  +  b  by  y  +  l. 

35.  2^-x'  +  2x^-x^  +  o^^-^  by  a;  +  l. 

36.  a;^  +  2a^-2a;*4-2a^-l  by  x-\-l. 

37.  a^  —  2  a^c  -I-  4  ac^  —  aa^  —  4  c^a;  +  2  ca^  by  a  —  aj. 

38.  a"^  -  6»  4-  c^  +  3  a&c  by  a^  ^  6^  _^  c^  +  aft  -  ac  +  6c. 

39.  a;"  +  2/"  by  a;  -h  2/  to  five  terms  of  the  quotient. 

40.  a^  —  6a;H-5bya^  —  2a;  +  l,  using  detached  coefficients. 


PROCESS 


14-0  +  04-0  +  0-6  +  5 
1-2  +  1 


2  +  1 


1+2+3+4+5 


2- 

-1  +  0 

2- 

-4  +  2 

3-2  +  0 

3-6  +  3 

4- 

-3- 

-6 

4- 

-8  +  4 

5- 

-10  +  5 

5- 

-10  +  5 

a^  +  2a?^  +  3ar^  +  4x-+5 


80  ACADEMIC  ALGEBRA 

Divide,  using  detached  coefficients  when  convenient : 

41.  aj»H-8a;-h7  by  a^  +  2a;  +  l. 

42.  a«  + 38  a +12  by  a +  2. 

43.  m^  —  19  m  —  6  by  m  +  2. 

44.  m^  -  32  m2  -  4  m  +  8  by  m  -  2. 

45.  a«  +  27a2-9a-10  by  a^-3a  +  5. 

46.  21ic^  -29a^-8a.'2-)-6a;  +  4  by  3a;-2. 

47.  2a;^-lla^  +  16a;2-12a;  +  9  by  2a;-3. 

48.  30a;*-62aj3  +  60ic2-36a;  +  8  by  5a;-2. 

49.  27ic*-33a^  +  46a^-119a;  +  55  by  9ic-5. 

50.  «^-2a;^-x'3-10a;-36  by  a;-2. 

51.  a;*  — 4.1^  +  50^  — 4a;4-l  by  a^  —  a;  +  l. 

52.  al'-Q^-lOx'  +  Tx-^-W  hj  x'-2x-3. 

53.  2a;*  +  7ic«-27a^-8a;-f-16  by  a^  +  5a;-4. 

54.  28a;^4-6a^4-6a;2-6a;-2  by  4a;2  +  2a;  +  2. 

55.  7a^-6x*-\-2a^-x-2  hj  6x'-^5x  +  2. 

56.  25a;2_20a^  +  3a;*  +  16a;-6  by  3a^-8a;  +  2.  *" 

57.  Sx'-\-7ix^-\-6x'  +  3x-l  by  x'-{-x-\-l. 

58.  6x^-2Sx^  +  S0x'-lSx-\-4:hj  2x^-5x  +  2. 

59.  24a;^  +  32«3_i6a^_25a;-4  by  6r^-a;-4. 

60.  a^-2a;^  +  -JLiB3_f.  2a,2^_i^^,_^  5  by  a;-|. 

61.  a^-|a;44.29aj3_3ja.-2  +  fa;-i  by  a;-|. 

62.  a^-lx^^s^-l^a^^^x'-llx-\-^^^  by  a^-faj  +  f 

63.  |a^  +  ^2/3_^2;3_ia^2;  by  iaJ  +  i2/^-«• 
64. to  five  terms. 

1  +  a; 

65.  to  five  terms. 

1  —  a; 

66.  a^"'^  +  2/^""^^  by  a;"~^  +  y'*'*"^ 


DIVISION 


8x 


SPECIAL   CASES   IN   DIVISION 
110.   By  actual  division, 


x-y 
x  —  y 


=  x-\-y. 

=  a?-^xy-{-f. 


x-y 

^  —  'it 

--——=x^  +  a^y  +  3?^-  -{-xf-\-y*. 

From  the  above  we  infer  that  the  difference  of  the  same  powers 
of  two  numbers  is  divisible  by  the  difference  of  the  numbers. 


x  +  y 
a?  —  f 


x  +  y 


=  x-y. 

=.Q?  —  xy-\-y'^j     Rem.,  —  2^*. 


x-\-y 


=  op^  -  a^y -{- xy^  —  ?/. 

=  x*  —  a^y  H-  x^y^  —  xy^  4-  y*,     Rem.,  —2y^. 


I  x-\-y 

From  the  above  we  infer  that  the  difference  of  the  same  powers 
of  two  numbers  is  divisible  by  the  sum  of  the  numbers  only  when 
the  powers  are  even. 

a^  +  / 

a._y  =  »  +  y,     Rem.,  2 /. 

0^  +  2/" 


3. 


x  —  y 
x-y 


=  x^  -{-  xy  -{-  y^y     Rem.,  2  y^. 

=  a^-^a^y-\-xf-\-f,     Rem.,  2y*. 

^x^  +  x'y^  a^/  +  ^^  +  y\     I^em.,  2f. 


x-y 

From  the  above  we  infer  that  the  sum  of  the  same  powers  of 
two  numbers  is  not  divisible  by  the  difference  of  the  numbers. 

ACAD.    ALG.  6 


s^ 


ACADEMIC  ALGEBRA 


fa^  +  r 


x-{-y 
af  +  f 

x-\-y 
a^  +  2/' 

x-\-y 
a^  +  / 

x-^y 


x  —  y,     Rem.,  2  y\ 

x^-xy-^-y"^. 

7?  —  x^y  -\- xy"^  —  if,     Rem.,  2  jf. 

x^  —  x^y  4-  x^y^  —  xi/''^  +  ^Z"*- 


Observe  that  the  sum  of  the  same  powers  of  two  numbers  is 
divisible  by  the  sum  of  the  numbers  only  when  the  powers  are  odd. 

111.  Hence,  when  w  is  a  positive  integer. 
Principles.  —  1.    x""  —  y^  is  always  divisible  by  x  —  y. 

2.  X"  —  ?/"  is  divisible  by  x-\-y  only  when  n  is  even. 

3.  a?"  +  2/"  is  never  divisible  by  x  —  y. 

4.  ic"  _|_  y^'  is  divisible  by  x  -\-y  only  when  n  is  odd. 

112.  From  §  110,  the  following  law  of  signs  may  be  readily 
inferred : 

When  x  —  y  is  the  divisor,  the  signs  in  the  quotient  are  plus. 
When  x  +  y  is  the  divisor,  the  signs  in  the  quotient  are  alternately 
plus  and  minus. 

113.  The  following  law  of  exponents  may  also  be  inferred : 
When  x^  ±  2/"  is  divided  by  x±y,  the  quotient  is  homogeneous, 

the  exponent  of  x  decreasing  and  that  of  y  increasing  by  1  in  each 
successive  term. 

114.  Proofs  of  preceding  principles. 

Principle  1 


or 


a;n. 

-r 

x-y 

xr>-i  4-  x»-^y  +  •.. 

1st  Rem., 

-r 

2d  Rem., 

a;n-2j,2  _  yn 

nth  Rem., 

Xn-nyn_  y 

x'^y-  -  y 

n 

»  =  t/"  —  y"  =  0 

DIVISION  83 

By  dividing  until  several  remainders  are  obtained,  it  is  found  that  the 
first  tei-m  of  the  first  remainder  is  x^-^y  ;  of  the  second,  x'^-^y'^ ;  of  the  third, 
a;n-3y8  J  of  the  fourth,  x/'-^y^ ;  and  consequently  of  the  nth,  x"-"y".  But 
x"-**  =  x^^  which,  §  103,  equals  1.  Therefore,  the  first  term  of  the  wth  re- 
mainder reduces  to  y". 

Since  the  second  term  of  the  nth  remainder  is  —  y»»,  the  entire  nth  re- 
mainder is  y"  —  y«,  or  0 ;  that  is,  there  is  no  remainder,  and  the  division  is 
exact. 

Therefore,  x*"  —  y^  is  divisible  hy  x  —  y  when  x  and  y  represent  any  two 
numbers  and  n  is  any  positive  integer. 


Principle  2 

x» 

-r 

-Xn- 

_x»»- 

x  +  y 

ajn-l  _  xn  -^f 

1st  Rem., 

-r 

-  x«- V 

1 
»-3y8 

2d  Rem., 

Xn-2y2  _  y, 

x"-V  -f-  X' 

3d  Rem., 

-  x^--^y*  -  y* 

4th  Rem., 

X'-V 

yn 

Suggestion. — Since  the  second  term  of  each  remainder  is  negative,  no 
remainder  can  reduce  to  0  unless  its  first  term  is  positive.  Show  for  what 
values  of  n  such  remainders  reduce  to  0  when  n  is  a  positive  integer. 

Prove  Principle  3. 

Prove  Principle  4. 

Examples 

Write  by  inspection  the  quotient  of 

6.    JiillllL'.  11. 

m  —n 

71  —  1 

8.    t^.  13. 

c  —  d 

9. 4+^^         14. 


10. f-  15. 

a  +  h  x-\-l  a4-2 


a» 

-6« 

a 

-b 

m' 

+  n« 

m 

-f-?i 

a^ 

-f 

X 

+  y 

r' 

-s' 

r 

—  s 

a» 

+  6» 

x'-d 

x-\-S 

a^-S 

a-2 

a;«-32 

x-2 

c«  +  27 

c  +  3 

a^  4- 128 

84  ACADEMIC  ALGEBRA 

16.  Find  five  exact  binomial  divisors  of  a^  —  x*. 

Solution 

(jfi  —  oifi  is  divisible  hy  a  —  x  (Prin.  1). 

««  —  ^6  is  divisible  hy  a  +  x  (Prin.  2). 

Since  a^  —  x^  =(^a^)^  —  («^)^  a®  —  x,^  may  he  regarded  as  the  difference  of 
two  odd  powers,  and  is,  therefore,  divisible  by  a^  —  x^  (Prin.  1). 

Since  a^  —  x^  =(a^y  —  (^J^)^)  a^  —  ofi  may  be  regarded  as  the  difference  of 
two  squares,  and  is,  therefore,  divisible  by  a^  —  x^  (Prin.  1). 

Since  a^  —  x^  =(a^)^ —{x^y,  a^  —  x^  may  be  regarded  as  the  difference 
of  two  squares,  and  is,  therefore,  divisible  by  a^  +  x^  (Prin.  2). 

Therefore,  the  exact  binomial  divisors  of  a^  —  x^  are  a  —  x,  a  +  x,  a^  —  x^, 
a^  —  o;^,  and  a^  +  x^. 

17.  Find  an  exact  binomial  divisor  of  a^  +  a^. 

Solution 

Since  a^  +  x^  =(a^y  -\-(x^y,  a^  +  ofi  may  be  regarded  as  the  sum  of  the 
cubes  of  a^  and  ic'^,  and  is,  therefore,  divisible  by  a^  +  x^  (Prin.  4). 

Find  exact  binomial  divisors : 

18.  a^  -  ml  24.  x^  +  a\  30.  a*  -  h\  four. 

19.  a^-ml  25.  ay^+h^\  31.  a«  -  1,  five. 

20.  6^  +  3^.  26.  o}-^+h\  32.  a^-6^six. 

21.  a^-a^  27.  a^^  _^  6^1  33.  a^^  -  ft^^*,  five. 

22.  c*4-w*.  *       28.  a3-27.  34.  a^^  —  6^^,  eight. 

23.  a^  +  6^  29.  a«  -  27.  35.  a^^  _  512^  nine. 

Equations  and  Problems 
115.    1.   Find  the  value  of  x  in  the  equation  bx  —  b^  =  cx  —  c^. 


. 

Solution 

bx-h^  =  cx-  c2. 

Transposing, 

bx  —  ex  =  b^  —  c^. 

Collecting  coefficients  of  a;. 

(6  -  c)x  =  62  _  c2. 

Dividing  by  &  -  c, 

b-c 

DIVISION  86 

2.  Find  the  value  of  x  in  the  equation  x  —  a?  =  2  —  a;x. 

Solution 
X  —  a^  =  2  —  ax. 
Transposing,  ax  -\-  x  =  a^  +  2. 

Collecting  coefficients  of  x,     (a  +  l)x  =  a^  ^  2. 

Dividing  by  a  +  1,  x  =  ^^^  =  a'^  -a  +  1  +— ^— 

a  4-  1  a-\-\ 

Solve  the  following  equations : 

3.  1  a  —  10  =  a^  —  ax-\-bx.       6.  ex  —  &  —  d^  -\-  dx  =  0. 

4.  x  —  l  —  c  =  cx—(?  —  c^.        7.  o?  —  ax  —  2ab  +  hx-\-h'^  =  0. 

5.  2m^  — ??ia;-t-na;  — 2n^  =  0.     8.  2n^  + 5?i -f-a;  =  n^  — nx— 2. 

9.  n^a;  —  3  m^n^  -\-nx-\-  3  m**  +  a;  =  0. 

10.  a^ic  -  a=^  4- 2a2  + 5a; -5a +  10  =  0. 

11.  ^ah-a^-2hx  =  2h^-ax. 

12.  9a2  +  4ma;  =  -(3aa;-16m2). 

13.  c2/-c*-2c3_2c2  =  2c-y  +  l. 

14.  a?/ —  2  62^  H- 3  cy  =  a  —  2  6 -f- 3  c. 

15.  z  +  6n*-4n3  =  l -3nz4-2w-7i2. 

16.  a;-362_19262c3-4cx  +  16c2a;  =  0. 

17.  863-1862-576-26a;  +  7a;  +  77  =  0. 

Solve  the  following  problems : 

18.  A  drover,  who  had  5  times  as  many  sheep  as  oxen  and  \  as 
many  oxen  as  horses,  sold  all  for  $  2300,  —  the  horses  at  ^  35  a 
head,  the  oxen  at  ^  25  a  head,  and  the  sheep  at  $  4  a  head.  What 
was  the  number  of  each  ? 

19.  A  man  paid  yearly  a  certain  amount  of  money  for  taxes 
and  twice  that  amount  for  improvements,  and  received  for  rent 
3  times  as  much  as  he  paid  out  for  improvements.  If  his  net 
gain  per  year  was  f  300,  what  were  his  taxes  per  year  ? 

20.  A  owed  B  a  certain  sum  of  money  and  C  twice  as  much. 
D  owed  A  3  times  as  much  as  A  owed  B,  and  E  owed  A  5  times 
the  sum  A  owed  B.  A  found  that  if  he  could  settle  with  them  all 
he  would  have  $  5000.     How  much  did  he  owe  B  and  C  ? 


86  AC  A  DEM  re  ALGEBRA 

21.  After  taking  3  times  a  number  from  11  times  the  number 
and  adding  to  the  remainder  7  times  the  number,  the  result  was 
12  less  than  117.     What  was  the  number  ? 

22.  A  merchant  failed  in  business,  owing  A  3  times  as 
much  as  B,  C  twice  as  much  as  A,  and  D  as  much  as  A  and  B. 
If  the  entire  debt  to  A,  B,  C,  and  D  was  $  28,000,  how  much  did 
he  owe  each  ? 

23.  At  a  certain  election  there  were  three  candidates  for  the 
office  of  mayor.  A  received  half  as  many  votes  as  B  and  4  times 
as  many  as  C.  If  the  total  vote  lacked  25  votes  of  being  2300, 
how  many  votes  did  each  receive  ? 

24.  Three  boys  together  had  140  marbles.  If  the  second  boy 
had  twice  as  many  as  the  first  and  half  as  many  as  the  third,  how 
many  had  each  ? 

25.  In  a  certain  school  of  600  students  there  were  twice  as 
many  Sophomores  and  3  times  as  many  Freshmen  as  Juniors, 
and  40  more  Seniors  than  Juniors.  How  many  students  were 
there  in  each  class? 

26.  Divide  25  into  three  parts  such  that  the  first  is  one  third  of 
the  second  and  5  greater  than  the  third. 

27.  A,  B,  and  C  divided  $40  so  that  for  every  $2  A  received, 
B  and  C  each  received  $ 3.     What  was  the  share  of  each? 

28.  Divide  $2200  among  A,  B,  and  C,  so  that  B  shall  have 
twice  as  much  as  A  and  $200  less  than  C. 

29.  Divide  $351  among  three  persons  so  that  for  every  dime 
the  first  receives  the  second  shall  receive  25  cents  and  the  third  a 
dollar. 

30.  A  man  gave  equal  amounts  of  money  to  a  school  and  to  a 
librarj^,  and  ^  the  same  amount  to  a  hospital.  If  to  all  he  gave 
$  28,000,  what  sum  did  he  give  each  ? 

31.  When  wheat  was  worth  85  cents  a  bushel,  oats  35  cents  a 
bushel,  and  corn  60  cents  a  bushel,  a  man  bought  a  quantity  of 
wheat,  oats,  and  corn  for  $67.  If  he  bought  twice  as  many 
bushels  of  oats  as  of  wheat,  and  also  three  times  as  many  bushels 
of  corn  as  of  wheat,  how  many  bushels  of  each  did  he  buy  ? 


REVIEW 


87 


REVIEW 
116.    Simplify: 

1.   a^-\-2aVxy  —  3mn-\-^mn  —  4:a^  —  5a'\/xy  +  Sa^-\-4:aVxy 
—  2  ran. 

-6xy^  +  7?  +  2Vx  +  x'y-^f-\--y/y-^x-2a^y  +  3xy'^  +  ^. 

3 .  (I  a  -  3  6c  +  ^  c  -  7  6)  -  (f  a  +  ^  6c  +  i  c  H-  3  6). 

4o  (ctV  —  4  ay  4-  4  6c  +  ax)  —  (6V  —  4  6?/  —  aa;  +  6c). 

5.  a^-(af-5a;V  +  10«^/-10ar^^H-5«i/^-2r'). 

6.  ^a-|a;-(|a-^a;)-(36--V-a5-|a)4-ia. 

7.  {a?  +  3ahj  J^3ay'){a'  -2ay  +  y^. 

8.  (0.-2"  +  2  x"!/"  +  /")  (ar^"  —  2  a;''?/'*  +  2/^")- 

9.  {\x^^)^xy  +  :^f){\a?-\xy  +  \f). 
10.  (.2a2-.8a  +  r.6)(.la2+.4a+.8). 

Expand : 


11. 

(y-3)(2/  +  4). 

21. 

{x  +  m){x-irm). 

12. 

(2/ +  7)  (2/ -8). 

22. 

(:j^  +  7^{x  +  1). 

13. 

Cv-l)(y  +  2). 

23. 

{x-l){l  +  x). 

14. 

(2/ -5)  (2/ -9). 

24. 

(x  +  3y)(x-2y). 

15. 

(2/  +  8)(2/-4). 

25. 

(a"+6-)(a«-6«). 

16. 

(m  —  ic)  (m  4-  x). 

26. 

(a"*  +  6")(a''-6"'). 

17. 

(m  +  a)  (m  —  6). 

27. 

(a  +  6  +  c)(a4-6-c). 

18. 

(a;  —  m)  (a;  +  n). 

28. 

{x  +  y  +  z)(x-y-{-z}. 

19. 

(ar'  +  a:)  (a;  +  2). 

29. 

(r  +  s-t)(s-t-r). 

20. 

(a^4-4)(a;^-3). 

30. 

(m  -\- n  —  p)  (m  —  n  -i-  p). 

31.  (a-6)(a  +  6)(a2H-62). 

32.  (l-a;)(l+^)(l  +  a:')(l  +  a^). 

33.  (l-a;)(l+a;)(l-a;)(l  +  a;). 

34.  (m  +  7i)  (m  +  7i)  (m  —  n)  (m  —  n). 


88  ACADEMIC  ALGEBRA 

35.    (a*-\-a^-{-a^  +  a-{-l)(a—l). 

37.  (oc^  —  x'^ -\- a^  —  x^ -{- X  —  1)  (x -{- 1). 

38.  (a^  +  2a*+4a3  +  8a2+16a  +  32)(a-2> 

Square 

39.  2x  —  Sy.  42.  100-5.  45.  a  +  &  +  c  +  d 

40.  x*  —  aa^.  43.  n'  —  irV.  46.  2  a  — 36  — 4c. 

41.  50  —  1.  44.  Ix  —  h'^y.  47.  a;"-^  —  2/ —  a^. 

Expand : 

48.  (5a-4y)(5a-32/).  51.  (2 a^o; -  5 5^) (4 a'a; - 3 6^)- 

49.  (6  a;  — 4?/)  (3  a; +  5  2/).  52.  (6  amn -h  5^9)  (6  amn  -  3p). 

50.  (3 a; -fa?/) (3 a; +  62/)-  53.  (3a"+i-^26^-i)(2a"+i- 3  6"-^). 

54.  (^x  +  y){x-y){7?-^y^){a^+f){a?  +  f). 

55.  (m«  4- 1)  (m^  + 1)  (m^  +  1)  (m  +  1)  (m  -  1). 

56.  (16a;*  +  l)(4a;2_|.i>)(2a;  +  l)(2a;-l). 
Divide : 

57.  ■x^-2x'^-ir2a?-\-\2x^-x-^hy  x-\-l. 

58.  x^-4.a?  +  6x^-^x  +  l  by  a^-3a;-f-l. 

59.  a;8-45a;5^45^4_j^8^_^93,_jL  by  a;3_  4jp2  ^  3^  _  ^ 

60.  a^-12a=^-a-f-12  by  a3-2a2-f4a-3. 

61.  6»-1062_55  +  4  by  63-262  +  35-1. 

62.  m^*'  -  6  m^  +  5  m  -  2  by  m*  4-  2  m^  -  3  m  -  2. 

63.  a^-160a4+127a3-100a2-20a+16  by  a3-6a2+5a-4. 

64.  6i«+29  6^-170 6^-61 62+210 6-22  by  6^+2  62-5  6-11. 

65.  6a^  +  ffa22/«- If  a2/«  +  |i/  by  a^  +  ^a?y  -  \af  ^  \f. 

66.  ah-ah'^+acd-<M'^-ahc+h^-hcd+hd^-ac^^-cb^-c^d+cd^ 
by  ac  —  6^  +  cd  —  d\ 


REVIEW  89 


67.    Divide  1  by  1  —  a;  to  six  terms. 

Simplify :  ' 

68     4a2  +  3  62  _  (2a2  -  3  62)  _  (7  2,2  _  2  a2)  _  (_  ^2  -  b^. 


69.    a'-  (b^  -  c2)  _  (62  4.  c2  _  a2)  +  (c^  -  6=^  -  a^)  -  (a^  -  6^  _  c^). 


70 .  x"  -  (2xy  -  f)  -  {a^  +  xy  -f)  -  ^  -2xy  -  y"  -\-ny\ 

71.  wt  +  52m-[n  +  3i)-(4j9-3n)-5n  +  2«i]-7i)j. 


72.  x-^y+\2z-(Py-4.x-lz)-(x-y)-z\-x. 

73.  1  - 51 -[ar^ -3- (2 ar^- 4) +3^2  + l]-(ar^- 4) j-1. 

74.  a  -  (2  6  +  5a) -2  6 -[3a -(66- 5a) -a]+ 12a. 

75.  a;-;m-[a;  +  (3m-2a;)  +  5a;-(2a:  +  m)]-2a;  +  mj 


76.    a:  -  15a; -[6a;  -  (7a;  -  8a; -9x)-10a;]+ 11  a;J  4- 9a;. 


77.  3 -Sc-[5- (2c -7 -3c- 11) +  5c]-6c-20i. 

78.  a;  +  J32/H-[4a;-(22/-7a;)-32/]-(10y  +  4a;)  +  8?/J. 

79.  l-;-[-(l-a;)-l]-l|-Ja;-(5-3a;)-7  +  a;J. 

Collect  the  coefficients  of  a;,  y,  and  z : 

80.  ax -\-  ay  -\- az  —  bx  —  by  —  bz. 

81.  aa;  — 2?/ -I- C2;  + 61/  — 12x4-42;. 

82.  3ma;  —  7ia;-|- 6y  —  y  +  3c2;  —  4^. 

83.  2^y  —  y  —  ^z-\-bz  —  x-{-  mx  —  71a;  —  z. 

84.  ca;  —  6?/  —  3  a2!  +  a;  —  2/  —  4  2  -f  2;  —  y. 

85.  16  ny  —  16  mx  -f  ax  -{- by -{- ex  —  2 y. 

86.  mx  -\- ny  -\- az  +  2  ax  —  2  my  -\- 2  nz. 

87.  a;  —  2/  —  aa:  4-  8  ma;  +aby  —  a^-^y^  +  z. 

88.  a^a;  +  6^2/  —  2  aa;  —  2  C2!  -f  c^z;  +  a;  +  2/  +  2;. 

89.  m^x  —  n^y  +  iv^y  —  n^x  —  2  mnx  —  2  mny  —  nh  4-  z. 

90.  4  {ax  —  by-\-cz)  —  2  (bx  —  ay  —  cz)  —  2(x  — y  -^  z). 


'Si^-^-iy  cKju^  (lA^  V 


FACTORING 


117.  The  numbers  that  multiplied  together  produce  a  given 
number  are  called  its  Factors. 

The  faptors  of  12  a  are  2,  2,  3,  and  a ;  or  4,  3,  and  a  ;  or  2,  6,  and  a  ; 
or  12  and  a  ;  or  2  and  6a;  or  4  a  and  3,  etc. 

118.  A  number  that  has  no  integral  factors  except  itself  and  1 
is  called  a  Prime  Number. 

119.  A  number  that  has  integral  factors  besides  itself  and  1  is 
called  a  Composite  Number. 

120.  A  factor  that  is  a  prime  number  is  called  a  Prime  Factor. 

121.  The  process  of  separating  a  number  into  its  factors  is 
called  Factoring. 

122.  To  factor  a  monomial. 

The  factors  of  the  numerical  coefficient  are  found  as  in  arith- 
metic, but  the  factors  of  the  literal  part  are  evident. 

Thus,  in  a^,  the  three  factors  are  as  evident  as  if  they  were  written  ax  ax  a. 

It  is  seldom  necessary  to  resolve  a  monomial  into  its  simplest 
factors,  but  the  following  problem  often  occurs : 

Given  one  factor  of  a  monomial,  to  find  the  other. 

Rule.  —  Divide  the  monomial  by  the  given  factor. 

1.  In  each  of  the  following,  if  xy  is  one  factor,  find  the  other: 
6  a^yy  15  ic*?/^,  2  a^y^,  a^oi^b^y^,   —  mnxy,  —  xy. 

2.  In  each  of  the  following,  if  abc  is  one  factor,  find  the  other : 
a^bc,  ab\  abc^,  —  a^^c^,   —  a^bc,   —  ^  abc. 

3.  Find  two  equal  positive  factors  oi  a^;  of  9  a^x^\  of  64  m*. 

4.  Find  two  equal  negative  factors  of  25  a;^ ;  of  16  a^  j  of  9  a^- 

90 


FA  C  TO  RING  91 

123.  To  factor  a  polynomial  whose  terms  have  a  common  factor. 

Examples 

1.  What  are  the  factors  of  3  a-xy  —  6  aoi^y  +  9  aooy^  ? 

PROCESS  Explanation. — By  examining  the  terms 

3  a^xu  —  6  ax^v  +  9  axi/^   ^^  ^^^  polynomial,  it  is  seen  that  3  axy  is  a 

=  3  axy  (a  —  2x  +  S  v)  factor  of  every  term.     Dividing  by  this  com- 

'^         mon  factor,  the  other  factor  is  found. 

Hence,  the  factors  are  3  axy,  the  monomial  factor,  and  (a  —  2  x  +  3  y),  the 

polynomial  factor,  since,  by  the  Distributive  Law  for  multiplication, 

Saxy(a-2x  +  Sy)  =  3  a^xy  -  6  ax'^y  -h  9  axy^. 

Find  the  factors  of  each  of  the  following  polynomials : 

2.  5x^—  5a^.  14.  ac  —  bc  —  cy  —  abc. 

3.  8ar^  +  2a;l  15.  Sa^f-SxY +  12  xy. 

4.  Sa^-Gx'y.  16.  16  a^ftV  -  24  a^ft-V  +  32  a^fc^c*. 

5.  4a2_6a6.  17.  60  m V/^ - 45  m»nVH- 90  mVr^. 

6.  5  m'  -  3  mil.  18.  12  a%  -  18  aby  +  24  a^by. 

7.  Sx^y^-Sx'f.  19.  14a%n2-21a«mV-49a%n2.. 

8.  5m*H-10mV.  20.  12  x^fs^  -  16  x'y'-z^  -  20  x'fz^. 

9.  4a=^6-6a262  gl.  25  c^da:^  +  35  (r'dV  -  55  c^dV. 

10.  5a;*-10a^-5ar^.  22.  51  a:?/'^^  -  68  a^^z^  _^  85  a;y«*. 

11.  Sa*-2a^b-{-a'b\  23.  52  a'b^c' -  65  a^b^c^  -  91  a'b^c^. 

12.  a;^2_^a;''  +  a;^o_a^.  24.  44  aV/ 4- 66  a V/ +  88  aV?^. 

13.  3  m^- 12  mV-h  6  mn^  25.  84a^2^-36ar'/+60iB2y-48a^. 

124.  To  factor  a  polynomial  whose  terms  may  be  grouped  to  show 
a  common  polynomial  factor. 

Examples 
1.    Factor  ax -\-  ay  -{- bx -^  by. 

Solution 

ax  +  ay  +  bx-\-by  =  a(x-\-y)-\-  bCx-^y) 
=  (a  +  6)(a;  +  2/). 


92 


ACADEMIC  ALGEBRA 


2.  Factor  ax  —  ay  —  hx  -\-  by. 

Solution 
ax  —  ay  —  hx  +  by 
=  a{x-y)-b(x-y) 
=  ia-b)(x-y). 

Observe  that,  when  the  first  two  terras  are  factored,  (x  —  y)  is  found  to 
be  the  binomial  factor.  Since  {x  —  y)  is  to  be  a  factor  of  the  other  two 
terms,  the  monomial  factor  is  —  6,  not  +  b. 

3.  Factor  ex  -\-y  —  dy  ^  Qy  —  dx-{-  x. 

Solution 

cx  +  y  —  dy  +  cy  —  dx-{-  x 
—  ex  —  dx  +  x  +  cy  —  dy  +  y 
z=(c-d-hl)x-\-(c-d+  l)y 
=  (c-d+l)(a;  +  2/). 


Arranging  terms, 


Factor  the  following : 

4.  am  —  an  -^  mx  —  nx. 

5.  be  —  bd -^  ex  —  dx. 

6.  pq  —  px  —  rq -\- rx. 

7.  ay  —  by  —  ab -\- b\ 

8.  a^  —  Qey—5x-\-5y. 

9.  b^  —  be-\-  ab  —  ac. 

10.  01^ -\- ocy  —  ax  —  ay. 

11.  c^  — 4  c  H- ac  —  4  a. 

12.  2x  — y-^  4x^  —  2 xy. 

13.  1  —  m  +  n  —  mn. 

14.  2p-\-q-\-6p^-^3pq. 

15.  ar  —  rs  ~  ab -\- bs. 


16.  x^-{-x^-\-x-{-l. 

17.  f^y2_Sy-S. 

18.  a^  +  ar^  H- ^2/ +  2/. 

19.  2-27i-n2  +  n«. 

20.  a^  —  a;  —  a  +  aic. 

21.  3a^-15a;  +  102/-2a^2/. 

22.  12a:'-Sab-3a'-i-2a^b. 

23.  3  m^Ti  —  9  mn^  +  ar/i  —  3  an. 

24.  15a&2~9  62c-35a6  +  216c. 

25.  16  aa;  4- 12  ay -8  6a;- 662/. 

26.  aa?^— aa;— aa^^Z+^y+ic— 1. 

27.  a;2/+aJ-3  2/^-3  2/-42/-4. 


28.  aa;  —  a  —  6a;  +  6  —  ca;  -|-  c. 

29.  mx—nx—x—my-{-ny-\-y. 

30.  a^— a— a6  +  6— 2ac+2c. 


FACTORING  93 

31.  mp^  —  np^  +  mq  —  nq -\- m  —  n. 

32.  aa?  —  hsr  —  (ix-{-hx-\-a  —  h. 

33.  2ma?  —  nx^-\-n-{-2nx  —  4:mx  —  2m. 

34.  hd(?  —  h  —  xy  —  y -\- ya?  —  hx. 

35.  a^x  —  a?y  —  ay  —  y -\- X -\- ax. 

36.  2-36  +  3a6-2a4-4a2_6a26. 

37.  m^ -^  mn -{- mn -\- n^ -{- m  +  n. 

125.  To  factor  a  trinomial  that  is  a  perfect  square. 

1.  Since  {a  -\-  h){a  -\-  h)  =  a^  -{-  2  ah  -\-  6^,  what  are  the  factors  of 
a^  4-  2  a6  -h  6^  ?     How  are  they  obtained  from  a^ -\- 2  ab -\- V"  ? 

2.  Since  (a  —  6)(a  —  6)  =  a'^  —  2  a6  4-  6*,  what  are  the  factors  of 
a^  -  2  a6  +  62  ?     How  are  they  obtained  from  a^-2ah-\-h^? 

3.  What  determines  the  sign  that  connects  the  terms  of  each 
factor  ? 

126.  One  of  the  two  equal  factors  of  a  number  is  called  its 
Square  Root. 

127.  In  a  trinomial  that  is  a  perfect  square  one  term  is  equal 
to  twice  the  product  of  the  square  roots  of  the  other  two  terms. 

26  a;2  —  20  ajy  +  4  y"  is  a  perfect  square,  for  twice  the  product  of  the  square 
root  of  25  x^  and  the  square  root  of  4  y^  is  20  xy. 

Rule.  —  Connect  the  square  roots  of  the  teiins  that  are  squares 
with  the  sign  of  the  other  tenriy  and  indicate  thai  the  result  is  to 
he  taken  twice  as  a  factor. 

From  any  expression  that  is  to  be  factored,  the  monomial 
factors  should  usually  first  be  removed. 

Thus,  2a3-4a2  +  2a  =  2  a{a^  -2a  +  l)  =  2a(a-  I)*. 

Examples 
Factor  the  following : 

1.  x^-Jr2xy-\-y\  5.  x^-{-6x  +  9. 

2.  p^-2pq-\-q^  6.  m'-8m  +  16. 

3.  c^-\-2cd-\-d\  7.  x^-2x-{-l. 

4.  m^-2mn-\-n\  8.  a^ -  16 a -{- 64:, 


94  ACADEMIC  ALGEBRA 

9.  a^-h4a;  +  4.  20.  9  +  42 6^ _|_ 49 56 

10.  ^  —  4:a-{-a\  21.  9m«  — 6m^  +  l. 

11.  4a-4a2-|-a3.  22.  A x^ - 20 xy  +  25. 

12.  Sx^-\-6xy-\-Sy\  23.  4: x^ -^  12 xyz -{- 9 yh^ 
'13.  2m2-4mn  +  2w2.  24.  9a2m2_6am  +  l. 

14.  5a^  +  30a;4-45.  25.  2 x -j- 20 a'x -\- 50 a'^x. 

15.  10  a^- 20  a; +  10.  26.  18  a^6  +  60  ad^  +  50  fe^. 

16.  16p2_24p  +  9.  27.  a'x^-2o.x^bf  +  by. 

17.  9iB2-42a;  +  49.  28.  25 .a^- -  60  «'"&''  + 36  fe^ 

18.  1+4  6  +  462.  29.  or^" - 2 a;"?/^^'*  + /V". 

19.  l-6a3  +  9a^  30.  81  a^dV  +  18 aft^c^d  +  ft^cS^l 

When  either  or  both  of  the  squares  are  polynomials,  the  expres- 
sion may  be  factored  in  a  similar  manner. 

31.  Factor  a^-\-6x(x  —  y)-{-9(x  —  yy. 

Solution 

=  [x  +  3  (x  -  ?/)][x  +  3  («  -  j^)] 
=  (x  +  Sx-Sy)(x  +  Sx-3y) 
=  (i4x-Sy)i4x-Sy). 

32.  Factor  (a-bf +  2(a-b)(b-c)  +  (b -cf. 

Solution 
(a  -  6)2  +  2  (a  -  6)(6  -  c)  +  (6  -  c)2 
=  [(«  -  6)  +  (6  -  c)][(a  -  &)  +  (6  -  c)] 
=  (a-6  +  6-c)(a-6  +  6-c) 
=  Ca  —  c)(^a  —  c). 


Factor : 


33.  x^  +  2x(x-y)  +  (x-yy. 

34.  a2_4^(^_;^^)_^4(^_l)2 

35.  c2-6c(a-c)  +  9(a-c/. 

36.  m2  +  2m(m  — ri)  +  (m  — n)2. 

37.  16-24(a-6)+9(a-6)2 

38.  x'-\-25(f-xy-\-10x(y-x). 

39.  14a(a;-2/)  +  (a;-2/f +  49al 

40.  10m(m-4)  +  25m2  +  (m-4)2 


FACTORING  95 

41.  (a-f  &)'-2(a  +  6)(6  +  c)  +  (6-hc)2. 

42.  {a  -  2  xf  -{-  4:{a  -2  x)(2  X  -h)  -\-  4.{2x  -  h)\ 

43.  16(a  -  xf  +  32(a  -  »)  (a;  +  6)  +  16(a;  +  hf. 

44 .  (a  +  3  5)2  -  4  (a  4-  3  6)  (3  6  -  2  c)  -f-  4  (3  6  -  2  c)l 

45.  (ar^  +  a;  H-  1)^+  2(ar  -f  1) (a;^  _^  ^  ^  l^^  _^  (^  _^  ly^ 

46.  (a  4-  &  +  c)2  +  2 (a  +  6  -  c)  (a  +  6  +  c)  +  (a  H-  6  -  c)«, 

47.  (a^-x2)2  +  2(aj3-a:2)(a;  +  l)  +  (a^H-2a;-f  1). 

128.    To  factor  the  difference  of  two  squares. 

1.  Since  {a-{-b){a—h)  =  a^—W,  what  are  tlie  factors  of  a^  —  h^? 
How  do  these  two  factors  differ  ? 

2.  Since   {o?  -\-  b^  (a^  —  W)  =  a*  —  6^,  what  are  the  factors   of 
a*  —  6*  ?     How  do  these  two  factors  differ  ? 

Rule.  —  Find  the  square  roots  of  the  tivo  terms,  and  make  their 
sum  one  factor  and  their  difference  the  other. 

Sometimes  the  factors  of  a  number  may  themselves  be  factored. 

Examples 

1.  Factor  b^  -  y\ 

Solution.  6"^  —  |/^  =  (6  +  y)  (ft  ~  y). 

2.  Factor  x^-1. 

Solution.  x'^  —  1  =(^x  +  l)(x  —  1). 

3.  Factor  x*  —  1. 

Solution.  x*  -  I  ={x^  +  1)(x^  -  1) 

=  (x-^  +  l)(x+l)(a;-l). 

Resolve  into  theix  simplest  factors  ; 

4.  x'-m^  9.    25 -c2.  14.  a'^-b^ 

5.  a^-y\  10.    a;^-49.  15.  4:x'-25y\ 

6.  a^~W.  11.    ic^-Sl.  16.  9a2_49  52 

7.  2/'-ar^.  12.    a*  -  16.  17.  a V  -  4  c^. 

8.  a^-9.  13.    a* -6*.  18.  m*  -  16  w^ 


96  ACADEMIC  ALGEBRA 

19.25  0^-1.  25.  400  a^- 100  2/^  31.5a^-5. 

20.  81m*-l.  26.  2a^-2h\  32.    3a^-3a. 

21.  ZQa'^-25.  27.  4m^-46^  33.    o?  -  xy\ 

22.  12162_c2.  28.  ^d'-Zf.  34.    5a^y-5ay. 

23.  400  a'' -81/.  29.  5x*-5y^\  35.    x"^  -  y""'. 

24.  100a^"-l.  30.  8a;i0-8/.  36.    a^^+i  -  a;?/-^ 

When   either   or   both   of  the   squares   are   polynomials,   the 
expression  may  be  factored  in  a  similar  manner. 

37.  Factor  25  a^  -  (3  a  +  2  b)\ 

Solution 
One  factor  is  5  a  +  (3  a  +  2  6),  and  the  other  is  5  a  —  (3  a  +  2  6). 

5a+(3a  +  2&)=5a  +  3a  +  26  =  8a  +  26  =  2(4a  +  6). 

5a-(3a  +  2&)=5«-3a-25  =  2a-2&  =  2(a-6). 

.-.  25a2  -(3  a  +  2  6)2  =(5  a  +  3  a  +  2  6)(5  «  -  3  a  -  2  6) 

=  (8a  +  26)(2a-26) 

=  2(4a  +  6).2(a-6) 

=  4(4a  +  6)(a-6). 
Factor 

38.  a2-(a  +  6)2.  42.  ^^-ia-xf. 

39.  62  _  (2  a +6)2.  43.  9  a^- (2  a -5)1 

40.  a2-(6  +  c)2.  *  44.  x"" -{Zx^ -2yy. 

41.  4c2-(64-c)2.  45.  49a2_(5a-46)2. 

46.  Factor  (3  a  -  2  6)^  -  (2  a  -  5  hf, 

*  Solution 

(3a -2  6)2 -(2a -56)2 
=  [(3  a  -  2  6)  +  (2  a  -  5  6)][(3  a  -  2  6)  - (2  a  -  5  6)] 
=  (3a-26  +  2a-5  6)(3a-26-2a  +  5  6) 
=  (6a-7  6)(a  +  3  6). 
Factor : 

47.  (2a  +  36)*-(a  +  6)2.  49.    (2  a;  +  5)^  -  (5  -  3  a;)2. 

48.  (5a-36)2-(a-^6)2.  50.    (a  -  2  6)^  -  (a  -  5)2. 


FACTORING  97 

51.  {2x-Zyf-{Zy^zf.  54.    (9  a;  +  6  2^)^  -  (4  «  -  3  ^)2. 

52.  (56-4c)2-(3a-2c)2.         55.    (a^  +  o^^ _ (2 a;  +  2)^. 

53.  i4.x-^yf-{2x-^af.       56.    (a  + 6 +  c)2-(a-6 -cf. 

57.  Factor  a^  +  4  —  c^  —  4  a. 

Solution 
a2  +  4  -  c2  -  4  a 
Arranging  terms,  =a2  —  4a  +  4  —  c^ 

=  (a-2)2_c2 
=  (a-2 +  c)(a-2 -c). 

58.  Factor  a*  -h  6^  -  c^  -  4  -  2  a6  +  4c. 

Solution 

a2  +  62_c2_4_2a6  +  4c 

Arranging  terms,  =  a^  —  2  aft  +  6*  _  ^2  _^  4  c  —  4 

=  (a2  -  2  a6  +  ft^)  _  (gs  _  4  c  +  4) 

=  (a-6)2-(c-2)2 

=  (a  -  6  +  c  -  2)(a  -  6  -  c  +  2). 
Factor : 

59.  a''-2ax-\-x^-n\  67.  c^  -  a^  -  &2  _  2a&. 

60.  h'^  +  2hy  +  y--n\  68.  6^  -  a:^  _  ^2  _^  2  a?y. 

61.  l-4g'  +  4^2_^2  gg  4c2-ar'_2/2-2a^. 

62.  ?^-2ra;4-aJ^-16^l  70.  9  c^  -  a^  -  y^  4.  2  a;y. 

63.  9a25_6a62_^63_45g2  ^^^  a:^  -  a^a;  -  4  ft^a;  -  4  afta;. 

64.  4a2c  +  12a5c  +  962c-4c3.  72.  dc^- 9a26  _  53  _  5^52 

65.  3a^y-12a;2/'  +  12  2/^-3a:V  73.  aW  -  4. a^ -12  a?c- 9 a&. 

66.  4an*-16aV+16a3-4a/i''.  74.  27c^-12a2  + 36a6  -  27  62. 

75.  a2-2a6-f  6^-c2  +  2ccZ-d2. 

76.  a^-2a;2/H-2/^-m2  4-10m-25. 

77.  4ar^4-9-12a;  +  10m7i-m2-25w*. 

78.  Q^-a:'  +  y''-h^  +  2xy-2db. 

ACAD.    ALO.  7 


98  ACADEMIC  ALGEBRA 

129.  To  factor  a  quadratic  trinomial. 

{x  +  'd){x-\-  5)  =  a^+  8X-I-15. 
{x-3){x-  5)=a^-  8  a; +  15. 
(x -f  l)(a;  + 15)  =  a^  +  16a;  +  15. 
(x  -  1)  (x  +  15)  =x^  +  14.x-  15. 
(x-3){x+  5)  =  r^+  2x-15. 
(x  +  S){x-    5)  =  x^-    2x-W. 

1.  How  may  the  first  term  of  each  factor  be  found  from  the 
product  ? 

2.  When  the  last  term  of  the  product  has  the  sign  -f,  how  do 
the  signs  of  the  last  terms  of  the  factors  compare  ?  How,  when 
the  last  term  of  the  product  has  the  sign  —  ? 

3.  How  does  the  coefficient  of  the  middle  term  of  the  product 
compare  with  the  algebraic  sum  of  the  last  terms  of  the  factors  ? 

130.  A  trinomial  of  the  form  o?  -\-hx-\-  c,  in  which  a?  is  the 
square  of  any  number,  c  the  product  of  two  numbers,  and  h  their 
algebraic  sum,  h  and  c  being  either  positive  or  negative,  is  called 
a  Quadratic  Trinomial. 

Rule.  —  Arrange  the  trinomial  according  to  the  descending  powers 
of  one  of  the  letters. 

For  the  first  term  of  each  fccctor  take  the  square  root  of  the  first 
term  of  the  trinomial;  ayid  for  the  second  terms,  such  numbers  that 
their  product  is  the  third  term  of  the  trinomial,  and  their  algebraic 
sum  multiplied  by  the  first  term  of  the  factor  will  be  equal  to  the 
second  term. 

Examples 

1.   Eesolve  a^  —  IScc  —  48  into  two  binomial  factors. 
Solution.  —  The  first  term  of  each  factor  is  evidently  x. 
Since  the  product  of  the  second  terms  of  the  two  binomial  factors  is  —  48, 
the  second  terms  must  have  opposite  signs ;  and  since  their  algebraic  sum, 

—  13,  is  negative,  the  negative  term  must  be  numerically  larger  than  the 
positive  term. 

The  two  factors  of  —  48  whose  sum  is  negative  may  be  1  and  —  48,  2  and 

—  24,  3  and  —  16,  4  and  —  12,  or  6  and  —  8.     Since  the  algebraic  sum  of  3 
and  —  16  is  —  13,  3  and  —  16  are  the  factors  of  —  48  sought. 

.-.  a;2  -  13x  -  48  =(x  +  3)(x  -  16). 


FACTORING  99 

^  2.    Factor  m^  -\-m  —  72. 

Solution.  —  Since  +  9  and  —  8  are  the  only  two  factors  of  —  72  whose 
algebraic  sum  is  +  1,  the  coefficient  of  the  middle  term,  the  required  factors 
are  (m  +  9)  and  (m  —  8). 

.-.  m2  +  m  -  72  =(m  +  9)(m  -  8). 

Separate  the  following  into  their  simplest  factors : 

3.  x^  +  lx  +  12.  18.    ar^  +  5  aa;  +  6  a^. 

4.  y^-ly-^12.  19.    aj2  -  6  aa;  +  5  a^. 
6.  y_82)-+:12.                         20.   y^ -Aby  -  12  b\ 

6.  7^-\-Sr-\-12.  21.  f-Sny-2Sn\ 

7.  m^  + 5  m  — 14.  22.  z^  —  anz  —  2ahi\ 

8.  a2-2a-15.  23.  x*  +  19  cx^  _^  90  ^^ 

9.  62_|_5_i2.  24.  a:«  + 12 aa;^  + 20 a^. 

10.  r'^r -30.  25.  a;^^ -  11  6V  +  24 6*. 

11.  c^-c- 72.  26.  5nx^-55nx  +  150n. 

12.  c2-5c-14.  27.  Sa^bx'-Sa'bx-Ga'b. 

13.  a;2-a;-110.  28.  12  mV  -  60  m^fta;  +  72  m^ft^. 

14.  a2  +  9a-52.  29.  4 aa;  +  2 oar' - 48 a. 

15.  (1^  + 8a -128.  30.  11  a^a; - 55 aa;  +  66 «. 

16.  ar^- 25  a; +  100.  31.  20  6x -h  10  6^  -  630  ar^. 

17.  a^  +  12a;-85.  32.  a^ -\-(b  -  a)x  -  ab. 
131.   To  factor  trinomials  of  the  form  ax^  -\-  bx  -{-  c. , 

Examples 

1.    Separate  3  a:^  +  11  a;  +  6  into  two  binomial  factors. 

Solution.  —  Since  3  x^  is  the  product  of  the  first  terms  of  the  factors,  and 
6  is  the  product  of  their  last  terms,  and  since  the  only  factors,  each  contain- 
ing X,  that  3  x2  can  have  are  3  x  and  x,  one  factor  is  3  x  plus  a  factor  of  6,  and 
the  other  is  x  plus  the  other  factor  of  6.  By  trial,  it  is  found  that  2  and  3 
are  the  factors  of  6,  and  that  if  2  is  added  to  3  ic  and  3  to  x,  the  middle  term 
of  the  product  (3  ic  +  2)  (x  +  3)  will  be  11  x. 

.-.  3x2+ llx  +  6=(3x  +  2)(a;  +  3). 


100  ACADEMIC  ALGEBRA 

2.  Factor  9  ic^  ^  30  a;  +  16. 

Solution 
9  x2  +  30  X  +  16 
=  (3a;)2+10(3a;)+16 
Put  w  for  3a;,  =m2+10m  +  16 

§130,  =(m  +  2)(m  +  8) 

Put  dxiovm,  =(3 X  +  2)(3 a;  +  8). 

Suggestion.  —  When  the  coefficient  of  x^  is  a  square^  and  when  the  square 
root  of  the  coefficient  of  oj^  is  exactly  contained  in  the  coefficient  of  x^  the 
trinomial  may  be  factored  by  the  method  illustrated  above. 

3.  Factor  4a;2  — 5  a;  — 6. 

Solution 

4x2  —  5a;  —  6=(4a:2  —  5a:  —  6)x-  = 

4  4 

_(4x)2-5(4x)-24^(4x-8)(4x  +  3) 
4  4 

=  4C^-^)(^^  +  ^)=(a;-2)(4x  +  3). 
4 

Explanation.  —  Although  the  first  term  is  a  square,  its  square  root  is  not 
exactly  contained  in  the  second  term. 

But  if  such  a  trinomial  is  multiplied  by  the  coefficient  of  a;2,  the  resulting 
trinomial  will  be  one  whose  second  term  exactly  contains  the  square  root  of 
its  first  term. 

Multiplying  the  given  trinomial  by  4,  factoring  as  in  example  2,  and  divid- 
ing the  result  by  4,  the  factors  of  the  given  trinomial  are  (x— 2)  and  (4x+3). 

4.  Factor  24.0?  +  14.x- ^. 

Solution 

24««+ 14x  _  5  =(24^^  +  Ur  -  6)x  §  =  lM^i8i£^^ 

6  6 

^  (12  x)2  +  7(12  X)  -  30  ^  (12  X  -f  10)  (12  X  -  3) 
6  6 

^2(6x  +  5).3(4x-l)^^ex  +  5)(4x-l). 

2i  Y.  o 

Suggestion.  — When  the  first  term  is  not  a  square^  it  may  always  be  made 
a  square  whose  square  root  will  be  exactly  contained  in  the  second  term  by 
multiplying  the  trinomial  hy  the  coefficient  of  x^. 


FACTORING  IGl 

After  factoring,  the  result  should  be  divided  by  the  coefiBcient  of  x'^. 
Frequently  the  multiplier  can  be  taken  smaller  than  the  coefficient  of  x% 
\  in  the  above  example. 

Separate  into  their  simplest  factors : 

5.  2ar^-}-a;-15.  17.  9»*-10a^-16. 

6.  9a:2-42a;  +  40.  18.  276^-362-14. 

7.  5ar^  4- 13a; +  6.  19.  10  iB«  -  2 a.-' -  44. 

8.  ^x'-llx  +  lO.  20.  2»2H-5a??/  +  2/. 

9.  25a^-\-15x-^2.  21.  2a^ -\-3xy -2f. 

10.  16x'  +  20x-66.  22.  Sx'-lOxy-^Sf. 

11.  36ar'-48a;-20.  23.  6«2_j^-i^_35 

12.  9a:2^433._j^()  24.  6a^-lSx-\-6. 

13.  25a:2^25a;-24.  26.  15 ar^  ^  14 x  -  8. 

14.  49a2_42aj_55.  26.  ISar^H- 17  a; -4. 

15.  16a;2_^50a._21.  27.  21a*-a-10. 

16.  4a;2_4a,_35  28.  18a;2_33._3g 

132.   To  factor  the  sum  of  the  same  odd  powers  of  two  numbers. 

Examples 

1.  Factor  a^  +  6^. 

Solution 

§§  111-113,  a'  -{-bT  =(a-{-  6)(a«  -  a^b  +  a*b^  -  a^b^  +  a^b*  -  ab^  +  ¥) 

2.  Factor  m*-h32ar'. 

Solution 

w^  +  32  a;6  =  m5  +  (2  xy 
§§  111-113,  =  (w  +  2  X)  (m*  -  2  mSa;  +  4  m2x2  -  8  mx^  +  16  «*). 

3.  Factor  a;^  +  /. 

Solution 

§§    111-113,  =  (X2  +  2/2)  (X*  -  X2y2  _|.  y4) 


.'  1^2  '  A'CA&EMIC  ALGEBRA 

Factor  the  following : 

4.  m^  +  n\  7.  o;^  +  1.  10.  a^  +  32. 

5.  a^  +  a^.  8.  r^ -{- s^.  11.  p^-^27. 

6.  a'-^b'.  9.  x^-{-y\  12.  x'*  +  128. 

133.   To  factor  the  difference  of  the  same   odd  powers   of  two 
numbers. 

Examples 

1.  Factor  a^  —  b^. 

Solution 
.     §§  111-113,  a^  -b''  ={a-  b)  (a^  +  a^b  +  a'^b^  +  a^b^  +  a^b*  +  ab^  +  66). 

2.  Factor  m^-32ar'. 

Solution 
§§  111-113,  m&  -  32  x6  =  ms  -  (2  xy 

=  (to  -  2  x)  (m*  +  2  m^ic  +  4  w^a^a  +  8  mx^  +  16  x*). 

3.  Factor  a^"  -  6^ 

Solution 
§§  111-113,   aio  -  65  =  (a2)6  _  55  ^  (^2  _  5)  (^s  4.  ^65  +  ^452  +  ^253  +  ^,4) , 

Factor  the  following : 

4.  a'-b\  7.  f-S.  10.  a^-.z/io. 

5.  a^-b^  8.  a^-32.  11.  a;«  -  ?/3. 

6.  f-aK  9.  a^-1.  12.  a;^  -  243. 

134.   To  factor  the  difference  of  the  same  even  powers  of  two 
numbers. 

Examples 
1,   Factor  a^-b^ 

First  Solution 
§§  111-113,  aS  -  ¥=(a-  b)(a^  +  a*6  +  a%^  +  a^b^  +  ab^  +  65) 
=  (a  -  6)  (a5  4-  05253  +  0^45  4.  ^54  +  ^352  +  2^) 
=  (a  -  6) [a2(a8  +  68)  +  ab(a^  +  6^)  +  b^a^  +  63)] 
=  (a  -  6)  (a2  +  a6  +  62)  (aS  +  ^3) 
§  132,  =  (a  -  6)  (a2  +  ab  +  62)  (a  +  6)  (a2  -  a6  +  62). 


FACTORING  103 

Second  Solution 
§  128,  a^-h^=  {a^y  -  {b^y  =  (a^  ±  b^)  (a^  -  b^) 

§§  132,  133,  =  (a  +  b)(a^  -  ab -{-  b^) (a  -  b) (a2  ^  ab -t  b^). 

In  each  of  the  above  solutions  a^  —  b^  is  first  separated  into  two  factors. 
Thus,  a^  —  b^  is  divisible  by  a  —  b  ;  also  by  a^—b^,  since  a^  —  b^={a^y^—(b^y^. 

When  the  even  powers  are  regarded  as  squares,  as  in  the  second  solution, 
the  process  may  be  regarded  as  factoring  the  difference  of  two  squares. 

Separate  the  followiug  into  their  simplest  factors : 

2.  x'^-f.  5.    x'-16.  8.    l-6«. 

3.  x^-1.  6.    x^-Sl.  9.    64-/. 

4.  a^-b\  7.    a' -625.  10.    1  -  a^. 


11.  p^-q^  17.  ?-^-s«.  23.  1+a^. 

12.  p^-\-q\  18.  7^-\-s^  24.  x  +  a^. 

13.  i^-g^.  19.  mJ  -n\  25.  32n-n^ 

14.  r^  — s*.  20.  m'  +  w^.  26.  m^  —  a^. 

15.  7''-s^.  21.  7/18  — w^.  27.  h^-drb\ 

16.  7'^  +  s^.  22.  a«  — 1.  28.  125  +  a^ 

135.   To  factor  by  the  Factor  Theorem. 

1.  If  5  X  =  0,  what  must  be  the  value  of  x? 

2.  If  5  (ic  —  3)  =  0,  which  factor  reduces  the  first  member  to 
zero ;  that  is,  what  value  of  x  reduces  it  to  zero  ? 

3.  Since  the  expression  5(a;—  3)  reduces  to  zero  when  x  —  S 
reduces  to  zero,  and  a;  —  3  is  reduced  to  zero  by  substituting  3 
for  X,  what  value  substituted  for  x  will  reduce  to  zero  evei^  expres- 
sion of  which  x  —  ^  is  a  factor  ? 

4.  How,  then,  may  it  be  discovered  whether  a;  —  3  is  a  factor 
of  a  given  expression  containing  x  ? 

5.  AVliat  value  of  x  will  reduce  the  expression  x*^  —  8  to  zero  ? 
What  factor,  then,  has  a:^  —  8  ? 

6.  What  vnlue  of  x  will  reduce  the  expression  x*  —  8  a;  +  7  \)o 
zero  ?     What  factor,  then,  has  a^  —  8  a;  -|-  7  ? 


5. 

aj*  -  16. 

6. 

x*  -  81. 

7. 

a'  -  625. 

y  previous  pr 

17. 

7-^  -  s\ 

18. 

7-«-hS«. 

19. 

mJ  -  n\ 

20. 

ttC  +  rH. 

21. 

m^  —  n^. 

22. 

a«-l. 

104  ACADEMIC  ALGEBRA 

7.  When  does  a  rational  integral  expression  containing  x  have 
the  factor  a;  -  1  ?   x-2?   x-S?   x-a? 

136.  Factor  Theorem.  —  If  a  rational  integral  expression  containiyig 
X  reduces  to  zero  when  a  is  substituted  for  x,  it  is  exactly  divisible 
by  X  —  a. 

Demonstration.  —  Let  D  represent  any  rational  integral  expression  con- 
taining X,  and  let  Z>  reduce  to  zero  when  a  is  substituted  for  x. 

It  is  to  be  proved  that  D  is  exactly  divisible  by  x  —  a. 

Suppose  that  the  dividend  D  is  divided  by  x  -  a  until  the  remainder  does 
not  contain  x.     Denote  the  remainder  by  B  and  the  quotient  by  Q. 

Then,  D  =  Q(x  -  a)-\-  B.  (1) 

But,  since,  by  supposition,  D  reduces  to  zero  when  x  =  a;  that  is,  when 
X  —  a  =  0,  equation  (1)  becomes 

0  =  0  +  22; 
whence,  B  =  0. 

That  is,  the  remainder  is  zero,  and  the  division  is  exact. 

Note,  a  is  the  known  number  substituted  for  x,  and  it  may  be  either 
positive  or  negative. 

Examples 

1.  Factor  a:^  —  8  ic^  + 17  a;  —  10  by  the  factor  theorem. 

Solution 

ic8-8x2  +  i7a:-10 

=  (a;-l)(x2-7x  +  10) 

=  (x-l)(x-2)(x-6). 

Since,  when  1  is  substituted  for  x,  x*  —  Sx^  +  17aj  —  10  reduces  to  0, 
X  —  I  is  a  factor  of  the  expression.  Dividing  by  x  —  1,  the  other  factor  is 
x2  —  7  X  +  10,  whose  factors,  §  130,  are  (x  —  2)  and  (x  —  5). 

2.  Factor  a;*  +  a^  —  16  a^  —  4  a;  +  48  by  the  factor  theorem. 

Solution 

X*  +  x8  -  16  x2  -  4  X  +  48 
=  (X  -  2)  (x3  +  3  x2  -  10  X  -  24) 
=  (x-2)(x  +  2)(x2  +  x- 12) 
=  (x-2)(x  +  2)(x-3)(x  +  4). 

Since,  when  2  is  substituted  for  x,  the  expression  reduces  to  0,  x  —  2  is 
a  factor  of  the  expression.  Dividing  by  x  —  2,  the  other  factor  is  x^  -|-  3  x^ 
—  10  X  -  24.  Since,  when  —  2  is  substituted  for  x,  x^  +  3  x2  —  10  x  —  24 
reduces  to  0,  x  +  2  is  a  factor  of  x^  +  3  x2  —  10  x  —  24. 


FACTORING  105 

Dividing  by  .r  +  2,  the  other  factor  is  x"^  -\-  x  —  12,  a  quadratic  trinomial 
whose  factors,  §  130,  are  (x  —  3)  and  (x  +  4).  Hence,  the  factors  of  the 
given  expression  are  (x  — 2),  (x  +  2),  (x  — 3),  and  (x  +  4). 

t 

Factor  the  following  polynomials  by  the  factor  theorem : 

3.  a:2_3ia;4.30.  21.  a^-7a;-h6. 

^  4.  4ar'-7a;  +  3.  22.  3?-l^x  +  S0. 

5.  26a^-10a;-16.  23.  a^-67a;-126. 

6.  48ar'-31a;-17.  24.  7?-^^x-10. 

7.  36ic2-61a;  +  25.  25.  a^  +  4a2 -  11a -30. 
-.    ^     8.  a^-9ar'  +  23a;-15.  26.  a^ +  9a2 +  26a +  24. 

9^5  ^9.  «'-13iB2  +  47a;-35.  27.  m^  -  Gm^  -  m-f  30. 

i^      10.  a^-14a^  +  35a;-22.  28.  6^-562-296  +  105. 

yil.  o^-4.x'-lx-^10.  29.  a^ +  10a2_  17 a -66. 

12.  a:3_6a^_9a;_j_l4.  30.  m3  + 7m2  +  2m -40. 

13.  3?-12(x?  +  ^lx-S0.  31.  63+166^  +  736  +  90. 

14.  ar'-lla;-^  +  31a;-21.  32.  ^i^  +  12  71^  +  41  w  +  42. 

15.  .^-10ar»  +  29a;-20.  33.  x*  -  15x2  + 10a;  +  24. 

16.  7?-l&a?^nx-m.  34.  a;*-25ar2  +  60x~36. 
T^  17.  a^-57aj  +  56.  V35.  x^  +  13  ar*  -  54  x  +  40. 
^            18.  x3-21aj  +  20.  36.  x*  +  22  a^  +  27  x  -  50. 

19.  a.-3-31a;-30.  37.    x^  -  9  ar^2/2  -  4  xi/S  +  12  ?/*. 

20.  aj3-13x  +  12.  38.   x^ -^x'y' +  12xif  -  4:^. 

39.  x'^-a?-lx^-\-x-\-Q. 

40.  x^-9a^  +  21a^  +  a;-30. 

41.  x*  +  8x3  +  14ar^-8a;-15. 

42.  x^- 4x^  +  19x2 -28a; +  12. 
1/43.  a^-18a:3  +  30x2-19a;  +  30. 

44.    a:«-10a;^  +  40a^-80a;2  +  80a;-32. 


106  ACADEMTC  ALGEBRA 

SPECIAL  APPLICATIONS  AND  DEVICES 

Examples 

137.  1.   Factor 

a2  -}-  62  ^  c2  4-  c?2  +  2  a6  -  2  ac  +  2  ac?  -  2  6c  +  2  6d  -  2  cd 

Solution.  —  Since  the  polynomial  consists  of  the  squares  of  four  numbers 
together  with  twice  the  product  of  each  of  them  by  each  succeeding  number, 
the  polynomial  is  the  square  of  the  sum  of  four  numbers,  §  95,  and  may  be 
separated  into  two  equal  factors  containing  a,  6,  c,  and  d  with  proper  signs. 

Since  —  2  ac,  —  2  6c,  and  —2  cd,  the  products  that  contain  c,  are  negative, 
while  2  a6,  2  ad,  and  2  bd,  the  products  that  do  not  contain  c,  are  positive, 
it  is  evident  that  the  sign  of  c  is  the  opposite  of  that  of  a,  6,  and  d. 

Therefore,  the  factors  are  either 

(a  +  b  —  c  -\-  d)  (a  +  b  -  c  +  d), 
or  (^— a —  b +  c  — d){- a  -  b  +  c  — d). 

Factor  the  following : 

2.  9x'-{-4.y'  +  25z'-12xy  +  30xz-207/z. 

3.  25m^-{-36n^+p^-60mn-10mp-i-12np. 

4.  a^-j-iex^  +  Setf-Saa^  +  Uay-ASx'y. 

5.  a^  -^  4:0^  -\-  b^  -\-  y^  -\-  4:ax  —  2  bx  +  2  xy  —  Aab  -\-  4:ay  —  2by. 

6.  m^  +  4  ^2  +  a^  -I-  9  —  4  mn  —  2  am  -f  6  m  +  4  ar^  —  12  ?i  —  6  a. 

138.  The  principle  by  which  the  difference  of  two  squares  is 
factored  has  many  special  applications. 

1.  Factor  a^'  +  a'b^  +  b^ 

Solution.  —  Since  a*  +  a^b^  +  6*  lacks  +  a%'^  of  being  a  perfect  square, 
and  since  the  value  of  the  polynomial  will  not  be  changed  by  adding  a'^b^ 
and  also  subtracting  a'^b'^,  the  polynomial  may  be  written 

a*  +  2  rt262  +  64  -  a262, 
which  is  the  difference  of  two  squares. 

a*  +  a^b^  +  6*  =  a*  +  2  a'^b'^  +  6*  -  a%^ 
=  (a2  +  62)2-(a6)2 
=  (a2  +  a6  +  62)  (a^  -  a6  +  62). 

2.  Factor  4a^-13a2-h9. 

Solution.  4  a*  -  ISa'^ -^9  =  4a'^  -  12  a^  +  9  -  a^ 

=  (2  a2  _  3)2  _  0,2 

=  (2  a^-\-a-  3)(2  a^  -  a  -  3). 


FACTORING  107 

3.   Factor  a*  +  4. 
Solution.  a*  +  4  =  a*  +  4  a^  _|.  4  _  4  ^52 

=  (a2  4.2)2_(2a)2 
=  (a2  +  2  a  +  2)  (a2  _  2  a  +  2). 
Factor  the  following : 

5.  a8  +  a*6*  +  68.  12.  ?i8  +  7i*  +  l. 

6.  p^+pV  +  g*.  13.  16  a^  + 4  3^2/2  + 2/4. 

7.  9a;*H-20a:2/  +  162/*.  14.  a^'h*  -  21  a^h^  +  ZQ. 

8.  4a4  +  lla26'  +  9  6^  15.  c^  +  c^dV  +  dV. 

9.  16 a*- 17 aV  +  a;!  16.  25 a* -  14 a^ft*  +  fts. 
10.  25a^-29a2/  +  42/*.  17.  9 a*  +  26 a^fes  _^  25 6*. 

18.  6*4-64.  21.    a* +  324.  24.    a;* +  64  2^-. 

19.  a* +  4 6*.  22.    a»-16.  25.    4  a* +  81. 

20.  m«  +  4.  23.    m*  +  4m7i*.  26.    a^y^ -\-4:xf. 

139.   Many  polynomials  may  be  written  as  quadratic  trinomials 
in  which  a?  and  x  are  replaced  by  polynomials. 

1.   Factor  9  ar^  +  4  ^^  _^  12  z^ -|- 21  a»  +  14  2/z  +  12  a^. 

Solution.  %  x'^  -\-  A  y"^  +  \2  z"^  +  2\  xz  -{-  14  yz  +  12  xy 

=  (9x2+  12  a;?/ +  4  2/2)  4- (21x2+  14^2:)+  12*2 
=  (3a;  +  2y)2  +  70(3x  +  2y)+40.3« 

§130,  =(3x  +  2y  +  4;s)(3x  +  2y  +  30). 

Factor  the  following : 

2.  a2  +  2a6  +  62  +  8ac  +  86c  +  15c2^ 

3.  a^-6a^  +  9/  +  6a^-182/2 +52;*. 

4.  m^  +  n^  —  2  77171  +  7  mp  —7np  —  SOp\ 

5.  16ri2  +  55-647i-1677i  +  m2  +  8m7i. 

6.  9m*  +  A:2- 30 +  397712 +  13  A; +  6m2A;. 

7.  25a2  +  2/2  +  10a.'2  +  10a2/-35aa;-7a^. 

8.  4a^  +  /  — 6^2  — 4a^  +  2a;2;  — 2/2:. 

9.  a2  +  62  +  c2  +  2a6  +  2ac  +  2  6c  +  5a  +  56  +  5c  +  6. 


108  ACADEMIC  ALGEBRA 

RBVIB^W  OP  FACTORING 
140.    Factor  the  following : 


1. 

m^-n\                 11. 

i>'  +  4. 

21. 

4  a;*  —  4  a;. 

2. 

x'-l.                    12. 

l-\-x^. 

22. 

7.v^-175. 

3. 

f  -  1.                     13. 

y- 

a\ 

23. 

8-27  a^a^. 

4. 

1  -  a^.                    14. 

^y- 

-/. 

24. 

32x-2a^. 

5. 

a^«-l.                   15. 

a}^- 

-  ah'K 

25. 

6b'-\-  24. 

6. 

x^-1.                    16. 

a'- 

256. 

26. 

a'  +  ^7  a\ 

7. 

1-a^                   17. 

3a' 

-3a. 

27. 

b'  - 196. 

8. 

1  -  b\                   18. 

64- 

-2^. 

28. 

450  -  2  a^. 

9. 

a-a\                   19. 

7n' 

-7n. 

29. 

4m3-h.004. 

10. 

b^  +  b.                   20. 

a'  - 

-9a. 

30. 

125-8a;«. 

31. 

i^-25x-{-100. 

45. 

y^-xy- 

42  a^. 

32, 

x'-xy-lS2y\ 

46. 

y?  —  ax  — 

72  a\ 

33. 

aay^  —Sax  — 4:  a. 

47. 

n^—  an  — 

90  a^. 

34. 

af^Bx'-ex. 

48. 

a?b^-\-ab- 

-56. 

35. 

3ar'  +  30a;  +  27. 

49. 

10  a^c  4- 33  ac- 7c. 

36. 

128  a^- 250  a*. 

50. 

60  7i/-61n2/-56M. 

37. 

5x^«  +  10a^-15. 

51. 

25a;2  +  60 

xy  +  36y\ 

38. 

6a^_19a;  +  15. 

52. 

6  ax^  4-  5  axy  —  6  ay^. 

39. 

a^n_^2af^yr'  +  fp. 

53. 

169a;*-26aaf +  a2iB2, 

40. 

7x'-77xy-S4:f. 

54. 

aV4-a'6'c2+6^ 

41. 

y^-2oyx-^136x'. 

55. 

lQx'  +  4.^f  +  y'. 

42. 

9i^-24:xy-{-16y'. 

56. 

b'c-13b' 

'c+42c. 

43. 

289a^-34a;2/  +  /. 

57. 

2a^-6ab-14:0b\ 

44. 

Sba^ +  bxy -10  by 

58. 

m^n  -  21 ', 

mn^  +  ^On^. 

FACTORING  109 

59.  17a^4-25a;-18.  74.  x^-{-a?y-41xy^-1^5f. 

60.  5  «2- 26x2/ +  5/.  75.  a;^  —  ca;  +  2  da;  —  2  c(?. 

61.  2/^  4- 16  a?/ -  36  CD^.  76.  a^y  +  4  a;^^  -  31  a;y  -  70  y. 

62.  8a2-21a6-96^  77.  a^  -  3aa;  +  46a;-12a6. 

63.  eOa'-^  +  Soaj-Sa^^^  78.  aa;^ -  9 aar^  +  26 aa; -  24 a. 

64.  30a;2-^37a;-77.  79.  12 ax -^hx-^ ay +  &hy. 

65.  2a:3_^28a,-2  +  66a;.  80.  25 «^ - 9 1/^ - 24 3/;^ - 16 ^2. 

66.  a^  +  W-c^-2db.  81.  x^-z^^y^-a^-2xy+2az. 

67.  aar^  +  10  aa;  —  39  a.  82.  2Wm—SaW+2hmx  —  3dbx. 

68.  n*  H- ?iW5^  +  a^6«.  83.  d'-^h^+(?-2ah-2ac+2hc. 

69.  aV  +  aV  +  a^.  84.  a^?/ + 14 ar'2/ +  43  a^  +  30 y. 

70.  a^-lQa-n.  85.  a^?/ -  1^  ^^^  +  38  a^  -  24  y. 

71.  aV  — 4aa;  +  3.  86.  a6a^  +  3  aftaj^  —  ofta;  —  3  a6. 

72.  6®  +  6y  +  2/*.  87.  3  6ma;  +  2  6?7i  — 3a7ia;— 2an. 

73.  a;7-2a;«  +  a;.  88.  20aa^-28aa:2_,_5^22._7^2 

89.  a;2^9y2_^2522_6a^-10a;2  +  302/2. 

90.  ^x^-\-y^  +  l^z^-Qxy-Syz-^24.zx. 

91.  ar'/22_|_^252_|_i_,_2rt^a;?/2  4-2a.'2/z  +  2(i6. 

92.  a262  -f  6V  +  c^d^  -  2  aft^c  +  2  a6cd  -  2  dc^d. 

93.  a:8  +  nV  +  ri»  +  27iV  +  2wV  +  2MV. 

94.  a^ftV  -  a252  _  52^^  4.  52  _  ^2^^  ^  ^2  _^  ^^  _  1, 

95.  (a  +  6)«-l.  100.  3a;«  +  96a;. 

96.  a«-2a2  +  l.  101.  (a  -  2)^  +  (a  -  1)'- 

97.  63-462  4-8.  102.  12a^4- 3a^- 8a;  -  2. 

98.  a;3_i0a;2_|.|25.  103.  2ar^  +  10a;  + aa;  + 5a. 

99.  8a;*-6a;2-35.  104.  a;^  4.  5  ^^^  _  29  a;  -  105 

105 .    m^n^  +  a^h^  -f  hhi^  +  2  hmn'  +  2  aft^^  +  2  a6mn. 


110  ACADEMIC   ALGEBRA 

106.  a'  +  a^b  +  a^b""  +  a'b^  +  ab'  +  6*. 

107.  a^b^-4:abx-4:X-\-2ab-\-4:a^. 

108.  (a  +  by(x  -y)-(a-j-b)(a^-  y^). 

109.  1  -  a;2  _|_  ^5^2  ^  ^^3  _  53,  _  ^^^ 

110.  a^  —  oc^  -\-  a^y  —  xy  -{-  m^y  —  xy^. 

111.  a^"-2+6y +  2  0?'*-%. 

112.  ^^l^x" +  1^x^-12^, 

113.  4(a6  +  cd)'-(a'  +  &'-c2-cZ2)2. 

114.  a^-a3^  125.  x^-\-4tx. 

115.  (a2  +  62_g2^^2_4^2^2  126.  ^-x'-x^  +  x?. 

116.  a^62  +  a26-12.  127.  (a  +  &)' -  (Z>  -  c)^ 

117.  x?-xy-x?y-\-y'^.  128.  3a6(a  +  6)  4-a^  + &^ 

118.  aj^  -  4  ajy-f  2x^-16/.  129.  (x  +  2/)^  +  (x  -  2/)^ 

119.  a^- 6* -(a +  6) (a -6).  130.  a^-{a^bf. 

120.  aj3-6a^  +  12a;-8.  131.  «* -  11 9 a;y  +  2/*. 

121.  1000iB3_27  2/3^  132.  m^-i-m^-mn-mn\ 

122.  (a  +  a^V-a;'.  133.  (a^  -  2/')^  -  (a^  -  a:?/)l 

123.  l  +  (a;+l)^  134.  x^-y^ -3xY(x' -  y'). 

124.  ab  —  bx*" -{- x^'y'^  —  ay"^.  135.  (a^+6a:H-9)2— (3^+5.^+6)2. 

136.  or^  +  (a  +  6  —  c)a^  +  (ab  —  ac  —  bc)x  —  a6c. 

137.  Factor  32  —  ic^  by  the  factor  theorem. 

138.  Factor  16  +  5  ic  —  11  x^  by  the  factor  theorem. 

139.  If  n  is  odd,  factor  a:"  —  a"  by  the  factor  theorem. 

140.  If  n  is  odd,  factor  x^  +  r"  by  the  factor  theorem. 

141.  Factor  ar^  —  6  fea;^  +  12  b'^x  —  8  6^  by  the  factor  theorem. 

142.  Discover  by  the  factor  theorem  for  what  values   of  yi, 
between  1  and  20,  «"  +  a"  has  no  binomial  factors. 


FACTORING  111 

EQUATIONS   SOLVED   BY   FACTORING 
141.    1.    Find  the  value  of  a;  in  a-^  +  1  ==  10. 


a^  H-  1  -  10 


FIRST  PROCESS  Explanation.  —  Transposing    the 

known  term  1  to  the  second  member, 
the  first  member  contains  the  second 
Or  =    9  power,  only,  of  the  unknown  number. 

a;.aj=3.3  .♦.  a;  =  3  Separating  each  member  into  two  equal 

factors, 

a;.ic  =  3.3    orx-a;  =  -3.-3. 


or  ic  •  fc  =  —3  — 3  .*.  X  =  —3 
.-.  a;  =  ±  3 


Since,  if  a;  =  3,  a; .  x  =  3  •  3,  and  if 
a;=—  3,  a;-a;  =  —  3.—  3,  the  value  of  x  that  makes  a;'^  =  9,  or  that  makes 
a;2  +  1  =  10,  is  either  +  3  or  -  3  ;  that  is,  a:  =  ±  3. 

Find  the  values  of  x  in  the  following  equations : 

2.  a^  +  3  =  28.  7.   a^-24  =  120. 

3.  3:2  +  1  =  50.  8.   ar^  +  ll  =  180. 

4.  a.-2-5  =  59.  9.   ar^  -  11  =  110. 

5.  a;2_7  =  29.  10.    x" -h^  =  a" -2ah. 

6.  3^2  +  3,^84  11.    a^-An^^m^-Amn. 

12.  Find  the  value  of  a;  in  a:^  _|_  ^  _  iq. 

Explanation. — The  first  process 

SECOND   PROCESS  is  given  in  example  1. 

^   ,   1  -j  Q  In   the   second  process,  all  terms 

are    brought    to    the    first  member, 

XT  —  J  =    0  which  is  factored  as  the  difference  of 

(a;  _  3)  (iB  -|-  3)  =    0    '  the  squares  of  two  numbers. 

o       /i       1  n  Since  the  product  of  the  two  fac- 

.*.   a;  —  3  =  0,  whence  a;  =  3  ,       .    ^  ^  ^v.       •  i  *    n 

'  tors  us  0,  one  of  them  is  equal  to  0. 

V)r  a;  +  3  =  0,  whence  a;  =  —3       This  gives  x-3  =  0  or  a;  +  3  =  0; 

.'.  a;  =  ±  3  whence  a;  =  3  or  x=  —  3;   that  is., 

X  =  ±  3. 

Solve  the  following  equations : 

13.  a:2  +  35  =  39.  16.    a:^  _  3^2  ^  q 

14.  ar2-50  =  50.  17.    a^- 4  6^  =  0. 

15.  ar^4-90  =  91.  18.   x^-dn^  =  0. 


■^12  ACADEMIC  ALGEBRA 

19.  ar'-21  =  4.  '  24.  32-a^  =  2S, 

20.  a^-56  =  S.  25.  65-0^  =  16. 

21.  »2_3a2  =  6a2.  26.  4.x^-Sb'=Sb\ 

22.  aj2  +  5&*  =  6  6*.  27.  ar^  +  25  =  25  +  m^ 

23.  a;2-40  =  24.  28.  aj^  -  30  =  2  (2  6^ -^  15). 

29.  Solve  a^  +  2  am  =  a^  +  m^. 

Solution 
a;2  +  2  am  =  a2  ^.  ^2, 

x^  =  a^-2  am  +  m2. 
ic.a;  =  (a  —  w)(a  —  w), 
Qg  aj -a;  =  — (a  —  m)  •  —  (a  —  w). 

.*.  a:  =  ±(a  — m). 

Solve  the  following  equations : 

30.  x'-(f  =  d^-2cd.  36.  a^-c2  =  36-12c. 

31.  a^-62  =  46c4-4c2.  37.  a^  -  4  6^  =  36  -  246. 

32.  a^-n^=6n  +  9.  38.  x^-a'  =  9-6a. 

33.  a^  +  10a  =  a2  +  25.  39.  a.-^  -  6^  =  4  -  4^2. 

34.  a^-a2  =  2a  +  l.  40.  x^  -  a^fe^  =  2  a6  +  1. 

35.  a^-m2  =  8m4-16.  41.  a:^  _  ^4  ^  54  _  2  ^252^ 

42.   Find  the  value  of  a;  in  a^  +  4  a?  —  45. 

FIRST    PROCESS  SECOND    PROCESS 

ar^  +  4a;  =  45  a^  +  4a;  =  45 

a2  +  4a;-45=    0  a^H-4a;  +  4  =  49 

(a;-5)(a;  +  9)=    0  (a;  +  2)(a;  +  2)=7  •  7  or  -7  • -7 

.-.  a;  -  5  =    0  .-.   a;  +  2  =  7  or  -  7 

or                       a;  +  9=    0  a;  =  7-2or-7-2 

.••  aj  =  5  or  —   9  .•.  a;  =  5  or  —  9 


FACTORING  113 

Explanation.  — The  explanation  given  for  example  12  will  serve  for  the 
first  process. 

In  the  second  process,  it  is  seen  that,  by  adding  4  to  each  member  of  the 
equation,  the  first  member  will  become  the  square  of  the  binomial  (x  -f  2) . 
Solving  for  (a;  +  2)  as  for  x  in  previous  examples,  x  +  2  =  ±  7  ;  whence 
a;  =  ±7-2  =  5  or  -9. 

Suggestion.  — In  the  following  examples,  when  the  coefficient  of  the  first 
power  of  the  unknown  number  is  even^  either  of  the  above  processes  may  be 
used  ;  but  when  it  is  odd,  the  first  process  is  simpler. 

Solve  the  following  equations : 

43.  a.'2-6a;  =  40.  62.  a^  +  4a;  +  3  =  0. 

44.  a^-8a;  =  48.  63.  a^  +  6a;-f8  =  0. 

45.  a?-~bx  =  -4:.  64.  a;2-9a;  +  20  =  0. 
"46.    ar^-7a;  =  18.  65.  ar^  + lla;  +  30  =  0. 

47.  a^-|-10a;  =  56.  66.  x'  +  x-lZ2  =  0, 

48.  a?-\-12x  =  2%.  67.  32  =  4a;  +  ar». 

49.  (c2-3a;  =  40.  68.  3x  =  88-ar^. 

50.  ic2-9ic  =  36.  69.  160  =  ar^-6a?. 
61.  ar^ 4- 11  a;  =  26.  70.  4.y  =  y^ -1^2. 

52.  ay'-12x  =  4:5.  71.  600  =  7/-10y. 

53.  2/'-152/  =  54.  72.  x" -{-16x  -  36  =  0. 

54.  /-21y=46.  73.  a^ +  15x- S4:  =  0. 

55.  x^-10x  =  96.  74.  ?/2-8y-84  =  0. 

56.  y^-20y  =  96.  75.  y^-2ay  +  a^  =  0. 

57.  2/2  +  122,^85.  76.  x" +  2  bx -h  b'- =  0. 

58.  2/'  +  42=:13y.  77.  a^  +  4aa.' +  4a2=  0. 

59.  7/-\-63  =  16y.  78.  ;32  _j_  22 «  +  121  =  0. 

60.  v--60  =  llv.  79.  ar-(a  +  &)a;  +  a6  =  0 

61.  2/^4-140  =  72?/.  80.  x^ -\- (c -[- d)x -^  cd  =  0. 

ACAD.    ALG. 8 


114  ACADEMIC  ALGEBRA 

81.  a^+(a  +  2)a;H-2a  =  0.  83.    x^ -(a  -  d)x  -  ad  =  0. 

82.  y^-(c-n)y-nc  =  0.  84.    x^ -{b  ■j-7)x +  7  b  =  0. 

85.  (2x-{-3){2x-5)-(Sx-l)(x-2)=l. 

86.  (2a;-6)(3a;-2)-(5a;-9)(a;-2)=4. 

87.  Solve  6x'-\-5x-21=0. 

Solution 
6  ic2  +  5  a;  -  21  =  0. 
Factoring,  §  1.31,  (2x  -  S)CSx -\- 1)  =  0. 

.-.  2  a;  -  3  =  0, 
or  3  X  +  7  =  0. 

•••  ^  =  f    or  -  f 
Solve  the  following  equations : 

88.  3a^  +  2a:-l  =  0.  93.    7a^  +  6a;-l  =  0. 

89.  5x^-^4.x-l=0.  94.    2v^-9v-S5  =  0. 

90.  3/ +  2/ -10  =  0.  95.    6?/2-22?/  +  20  =  0. 

91.  3/ -42/ -4  =  0.  96.    3a;2  +  13a;  -  30  =  0. 

92.  4?/2  +  9?/-9  =  0.  97.    4  a:^  _^  ^^3  3.  _  j^2  =  0. 

98.  Solve  the  equation  a^  —  2a^  —  5x-j-6  =  0. 

Solution 

x3- 2x2 -5a; +  6=0. 
Factoring  by  the  factor  theorem,  (a;  —  l)(a;  —  3)  (x  +  2)  =  0. 
X  -  1  =  0,  or  X  -  3  =  0,  or  X  +  2  =  0. 
.-.  X  =  1,  or  3,  or  —  2. 

Solve  the  following  equations : 
99.    a;3_i5ic2_^71a._105  =  0.  101.    a;^  -  12  a; -|- 16  =  0. 

100.    a;»-|-10a;2+lla;-    70  =  0.  102.    a;^  -  19a;  -  30  =  0. 

103.  a;^  +  a.-»-21a;2_^_^20  =  0. 

104.  a;^-7a;3_j_^_l_g3^_9Q^Q 

105.  x'+Sx^-x^-eSx  +  60=:0. 

106.  a;^-lla;^  +  45ar^-85a.'2  +  74a;-24  =  0. 


HIGHEST   COMMON   DIVISOR 


'   142.    1.  Name  all  the  numbers  that  will  exactly  divide  both 
a^  and  a^     Which  of  these  is  of  the  highest  degree  ? 

2.  What  is  the  highest  divisor  common  to  h"^  and  6"^?  to  y? 
and  X? 

3.  Since  the  highest  divisor  common  to  a^  and  a^  is  a^,  to  J/ 
and  h^  is  h"^,  and  to  x^  and  x  is  x,  what  is  the  highest  divisor  com- 
mon to  a'*6 V  and  a^h^x  ? 

4.  What  is  the  highest  common  divisor  of  36  a^h  and  90  ah^  ? 
What  prime  factors,  or  divisors,  are  common  to  36  a^h  and  90  a6^  ? 
How  may  the  highest  common  divisor  of  36a''6  and  90a?>-  be 
found  from  their  factors? 

143.  A  number  that  exactly  divides  each  of  two  or  more  alge- 
braic expressions  is  called  a  Common  Divisor  of  them. 

The  common  divisors  of  12  a-  and  ^a^  are  2,  a,  4,  a^,  2  a,  4  a,  2  a^, 
and  4a2. 

An  exact  divisor  of  an  expression  is  a  factor  of  it. 
Two  expressions  whose  only  common  divisor,  or  factor,  is  1  are 
said  to  be  prime  to  each  other. 

3  X  and  2  a  are  prime  to  each  other  ;  also  x  -\-  y  and  x  —y. 

144.  That  common  divisor,  or  factor,  of  two  or  more  algebraic 
expressions  which  is  of  the  highest  degree  is  called  their  Highest 
Common  Divisor,  or  Highest  Common  Factor. 

The  highest  common  divisor,  or  factor,  of  12  cfi  and  4  a'^  is  4  a^. 

The  abbreviation  H.  C.  D.  is  used  for  Highest  Common  Divisor. 

The  highest  common  divisor  in  algebra  corresponds  to  the  greatest  com- 
mon divisor  in  arithmetic.  But  there  would  be  an  inaccuracy  in  applying 
the  term  greatest  common  divisor  to  literal  numbers,  since  letters  may  repre- 
sent any  numbers,  as,  for  instance,  fractions. 

116 


116  ACADEMIC  ALGEBRA 

Thus,  if  a  =  ^,  then  a^  =  ^,  a^  =  ^,  and  the  higher  the  degree  of  the  Mteral 
number,  the  less  will  be  its  arithmetical  value.  Consequently,  it  is  inaccurate 
to  speak  of  a^,  the  highest  common  divisor  of  a^,  2a^b,  and  3a^,  as  their 
greatest  common  divisor,  because  a  may  represent  a  proper  fraction. 

145.  Principle.  —  The  highest  common  divisor,  or  factor,  of  two 
or  more  algebraic  expressions  is  the  product  of  all  their  common 
prime  factors. 

146.  To  find  the  highest  common  divisor  of  expressions  that  may 
be  factored  readily  by  inspection. 

Examples 
1.   What  is  the  highest  common  divisor  of  12  a*b^c  and  8  a^6V  ? 

FIRST    PROCESS  SECOND    PROCESS 

12a'b^c  =  3  X  2  X  2  X  aaaa  x  bb  x  c         12a*b^c  =4:a^b^c  xSa^ 
Sa^b^<^  =2  X  2  X  2  X  aa  X  bbb  X  ccc  Sa^feV  =Aa^b^c  x2b(? 


H.C.D.  =  2  x2  X  aax  bb  xc  =  4:a%'c     H.  C.  D.  =  4  a^ft^c 

Explanation.  —  Since  the  highest  common  divisor  of  the  expressions  is 
the  product  of  all  their  common  prime  factors  (Prin.),  and  since  the  only 
prime  factors  common  to  the  given  expressions  are  2,  2,  a,  a,  b,  6,  and  c, 
their  product,  4:a^b%  is  the  highest  common  divisor. 

Suggestion.  —  Frequently  the  work  may  be  abridged  by  grouping  com- 
mon factors,  as  in  the  second  process.  Since  3  a^  and  2  bc^  are  prime  to 
each  other,  4  a^bH  must  contain  all  the  common  factors,  and  be  the  highest 
common  divisor. 

2.    What  is  the  H.  C.  D.  oi  Sa^-Sxy^  and  a?  -  2  x^y  +  xy^' '! 

PROCESS 

Za?-Zxy^  =Sx(x-{-y)(x-y) 

a?  —  2  a?y -\- xy^  =    x(x  —  y)(x  —  y) 

.-.  H.C.D.  =    x(x-y) 

Explanation. — For  convenience  in  selecting  the  common  divisors,  the 
expressions  are  resolved  into  their  simplest  factors. 

Since  the  only  common  prime  factors  are  x  and  (x  —  y),  the  highest 
common  divisor  sought  is  their  product,  x(x  —  y)  (Prin.). 


HIGHEST  COMMON  DIVISOR  117 

Rule.  —  Separate  the  expressions  into  their  prime  factors. 

The  product  of  all  the  common  prime  factors,  eaxih  factor  being 
taken  the  least  number  of  times  it  occurs  in  any  of  the  given  expres- 
sions, is  the  highest  commoyi  factor. 

The  factors  that  enter  into  the  H.  C.  D.  can  often  be  selected  without 
actually  separating  the  expressions  into  their  prime  factors. 

3.   What  is  the  H.  C.  D.  of  Ba^c'-d  b^c^  and  aV  -  6V  ^  ^2yS 

-by? 

Solution 

5  a2c2  -  5  62c2  =  5  c\a^  -  b^) 

q2a;8  -  62a;8  _|.  ^2^8  _  fc2y8  =  (a;3  4.  y8)  ^q2  _  52) 

.-.  H.  C.  D.  =  a2  -  62 

Find  the  highest  common  divisor  of 

4.  lOa^f,  10a^2/3,  and  15xy*z. 

5.  70a«6«,  21  a*b\  and  35  a*b\ 

6.  8mV,  28  mV,  and  56  mV. 

7.  4:b^cd,  6  6V,  and  24  aftc^. 

8.  10(x-y)*^  and  15  (z  -  y)  (x  -  yf. 

9.  4:(a-^by(a-b)  and  b(a  +  by(a-by. 

10.  3  (a^  -  by  and  a  (a  -  ?>)  (a^  -  b'). 

11.  »2-2a?-15  and  ic2-a;-20. 

12.  x*  —  y*,  a^  —  y^,  and  a; +  2/. 

13.  a2  +  7a  +  12  and  a2-t-5a  +  6. 

14.  a^-\-f  a,nd  x^-\-2xy  +  y\ 
^\h.  a^-y?  and  a^-2ax^-^' 

16.  a^  -  52  and  a^  +  2  a6  -f-  62. 

17.  ic*  -h  aj^i/'^  H-  ?/*  and  3^  +  0^  +  ?/*. 

18.  a^  H-  2/^,  a^  +  ^,  and  oi^y  +  a^^. 

19.  a^  +  a26*+68  and  3a2-3a62  +  36*. 

20.  a?  —  01?,  a^-\-2ax-\-  0?,  and  a^  +  a^. 

21.  ax  —  y-\-xy  —  a  and  aa::^  +  a^y  —  a-y^ 

22.  a^6  —  &  —  a^c  +  c  and  a6  —  oc  —  5  4-  c. 


118  ACADEMIC   ALGEBRA 

23.  1  —  4:X^,   1-f  2ar,  and  4  a  — 16aa^. 

24.  (a-6)(6-c)  and  (c-a)(a2-62). 

25.  24:a^f  +  8ay'f3indSa^f-Sx'f, 

,  26.  6  a.-2  +  a;  —  2  and  2x^  —  llx-\-5.    - 

,  27.  16a^-25  and  20a^-9a;-20. 

28.  x^-6x-{-5  Sindoc^-5x^-\-7x-3. 

29.  a^  -  4  and  a.-^  -  lOo^  +  31  oj  -  30. 

30.  a;2  -  9  and  a^  -  12a^  +  41  a;  -  42. 

31.  aj3  —  4  a;  +  3  and  a^  +  a;^  —  37  a;  +  35. 

147.  To  find  the  highest  common  divisor  of  expressions  that  can- 
not be  factored  readily  by  inspection. 

1.  What  are  the  exact  divisors  of  ab?  Will  they  be  factors 
of  2  times  ab?  of  a  times  ab?  of  m  times  ab ? 

2.  If  a  number  is  an  exact  divisor  of  an  expression,  what  will 
be  its  relation  to  any  number  of  times  the  expression  ? 

3.  What  common  divisor  have  ax  and  ay?  any  number  of  times 
ax  and  ay,  as  m  •  aa;  and  71  *  ay? 

4.  If  two  numbers  have  a  common  divisor,  what  divisor  has 
their  sum  ?  their  difference  ?  the  sum  or  difference  of  any  num- 
ber of  times  the  numbers  ? 

5.  What  is  the  highest  common  divisor  of  2am(x-}-y)  and 
3bm(x-{-y)?  How  will  it  be  affected,  if  the  second  number  is 
multiplied  by  7  or  2;  ?  by  2  or  a  ?  How  will  it  be  affected,  if  the 
first  number  is  multiplied  by  5  ?  by  6  ? 

6.  By  what  numbers  may  one  of  two  expressions  be  multi- 
plied without  affecting  their  highest  common  divisor  ? 

7.  How  will  the  highest  common  divisor  of  2am(x  -\-y)  and 
3  bm  {x  +  y)  be  affected,  if  the  first  number  is  divided  by  2  ? 
by  a?  by  m  ?  by  (x-\-y)?  How,  if  the  second  number  is  divided 
by  6  ?  by  m  ? 

8.  By  what  numbers  may  one  of  two  expressions  be  divided 
without  affecting  their  highest  common  divisor  ? 


HIGHEST  COMMON  DIVISOR  119 

148.  Principles.  —  1.  A  divisor  of  an  expression  is  a  divisor 
of  any  number  of  times  the  expression.     Hence,  by  §  ^^, 

2.  A  common  divisor  of  two  expressions  is  a  divisor  of  their  sum, 
of  their  difference,  and  of  the  sum  or  the  difference  of  any  number 
of  times  the  expressions  ;  also, 

3.  The  highest  common  divisor  of  two  expressions  is  not  affected 
by  multiplying  or  dividing  either  of  them  by  numbers  that  are  not 
factors  of  the  other. 

Examples 

1.    Find  the  H.  C.  D.  of  a^-f  5x4-6  and  4a^  +  21ar^  +  30a;4-8. 

PROCESS 

ar^  +  5  »  -h  6)4  a^  +  21  ar^  +  30  a;  -f  8(4  a?  + 1 

4  a:^  +  20  a:^  ^  24  a; 

a?-\-    6a;  +  8 
a?-\-   5a;  +  6 

X  +  2)a:*  4-  5  a;  4-  6(a;  +  3 

a^  +  2a; 

3a;-}-6 
.-.  H.  C.  D.  =  a;  +  2.  3  a;  +  6 

Explanation.  —  Since  the  highest  common  divisor  cannot  be  higher  than 
a;2  +  5  x  +  6,  it  will  be  x"^  -\-  b  x  -{■  Q,  if  jc^  +  5  a;  +  6  is  exactly  contained  in 
4  as'*  +  21  a;2  +  30  X  +  8.  By  trial,  it  is  found  that  it  is  not  exactly  contained 
in  4  a;'^  +  21  x'^  +  .30  a;  +  8,  since  there  is  a  remainder  of  a;  +  2.  Therefore, 
a;2  +  5  X  +  6  is  not  the  highest  common  divisor. 

Since  x*^  +  5  x  +  6  contains  the  highest  common  divisor,  (4  x  +  1)  times 
x2  +  5x  +  6  will  also  contain  the  highest  common  divisor  (Prin.  1);  and 
since  both  4  x^  +  21  x2  +  30  x  +  8  and  (4  x  +  1)  (x2  +  5  x  +  6)  contain  the 
highest  common  divisor,  their  difference,  x  +  2,  must  contain  the  highest 
common  divisor  (Prin.  2).  Hence,  the  highest  common  divisor  cannot  be 
higher  than  x  +  2. 

X  +  2  will  be  the  highest  common  divisor,  if  it  is  exactly  contained  in 
x2  +  5  X  +  6,  since,  if  it  is  contained  in  x^  +  5  x  +  6,  it  will  be  contained 
in  any  number  of  times  x^  +  5 x  4-  6,  as  (4 x  +  l)(x2  +  5x  +  6)  (Prin.  1); 
and  in  the  sum  of  (4x+ l)(x2  +  5x  +  6)  and  x  +  2,  or  4x3  +  21x2  +  30x  +  8 
(Prin.  2).  By  trial,  x  +  2  is  found  to  be  exactly  contained  in  x^  +  5  x  +  6. 
Therefore,  x  +  2  is  the  highest  common  divisor  of  the  given  expressions. 


120  ACADEMIC  ALGEBRA 

2.   Find  the  H.  C.  D.  of  6  a^  +  33  a;  -  63  and  2  a^+11  ay'-x-SO. 


PROCESS 


3)6  a^  +  33  a; 


63 


2a^-hllx-21 


2  a^  +  11  a^  -      X 
2  aj3  +  11  aj^  _  21  a: 


30(a; 


10)20  a;  -  30 

2  a;-    3)2  ar' +  11  a;  -  21(a;  +  7 
20^-    Sx 


%  H.C.D.  =2a;-3. 


14  a;  -  21 
14  a;  -  21 


Suggestion.  —  Since  only  common  factors  are  sought,  factors  that  are 
not  common  to  the  given  expressions,  as  3  and  10,  may  be  rejected  from  any 
expression  before  it  is  used  as  a  divisor  (Prin.  3). 

3.   Find  the  H.  C.  D.  of 

2  a;3  +  5  ar^  -  22  a;  + 15  and  5  a^s  +  18  «2  -  33  a;  +  10. 


PROCESS 


2a?  + 5  a;2- 22  a; -I- 15)5^3  ^  13  a.2 
2 


33  a;  + 10 


lOa^-hSex'-    66  a; +  20(5 
10  a^  +  25  ar^  -  110  a;  +  75 
11)11  a;^+    Ux-55 
a^_j_      4a;—    5 


x'  +  4.x-5)2a^-^5x'-22x-\- 15(2  a;  -  3 
2af-\-Sx'-10x 


H.C.D.  =  a;2_^4^_5 


3a;2 
Sx" 


12  a;  + 15 
12  a;  +  15 


Suggestion.  —  When  the  first  term  of  the  divisor  is  not  contained  in  the 
first  term  of  tlie  dividend  an  integral  number  of  times,  fractional  quotients 
may  be  avoided  by  multiplying  the  polynomial  taken  for  the  dividend  by 
some  number  not  a  factor  of  the  divisor  (Prin.  3).  In  the  above  example 
the  simplest  factor  that  may  thus  be  introduced  is  2,  if  5  x*+18  oj^— 33  x+ 10 
is  taken  for  the  dividend ;  or  5,  if  2  x^  +  5  oj^  —  22  x  +  16  is  taken  for  the 
dividend. 


HIGHEST  COMMON  DIVISOR  121 

4.   Find  the  H.  C.  D.  of 
30  amic^  —  21  ama;  —  99  am  and  42  aftar*  +  33  a6a^  —  45  aftaj. 

PROCESS 

30  ama^  —  21  amx  —  99  am)  42  aboi?  +  33  a^ar'  —  45  afta; 
Reject  m  (Prin.  3).  Eeject  hx  (Prin.  3). 

30  aa:^  _  21  aa;  -  99  a)42  aa^  +  33  aa;  -  45  a 
Reserve  the  common  factor  3  a  as  a  factor  of  the  H.  C.  D. 

lOa^-    Ix-    ^S)Ux'  +  llx-    15 

7  5 


70ar'-49a;-231)70a.'2  +  55aj-    75(1 
70^2  _  49a,  _  231 


52)  104  a; +  156 
2a;+      3 
2a; +  3)10x2 -7a;-    33(5a;-ll 
.-.  H.C.D.  =  3a(2a;  +  3).  lOx'-lx-   33 

Suggestion.  —  Since  each  of  the  polynomials  contains  a  factor  not  found 
in  the  other,  these  two  factors  may  be  rejected  (Prin.  3).  Consequently,  m 
is  rejected  from  the  firet  polynomial,  and  hx  from  the  second. 

To  simplify  the  process  the  common  factor  3  a  is  removed  and  reserved  as 
a  factor  of  the  H.  C.  D. 

5.   Find  the  H.  C.  D.  of  9ar2-35a;  +  24  and  29  a;  -  8  ar*  -  15. 

PROCESS 

-8a;^  +  29a;-15 
9a;2_35a;  +  24 

ar^-    Qx-^   9)9a;2_  35^.^24(9 
9a;2_54a;_|_81 

19)  19  a; -57 

X-    3)a;2-6a;  +  9(a;-3 
.-.  H.  CD.  =  a;  -  3.  a;^-6a;  +  9 

Suggestion.  —  Since  the  sum  or  the  difference  of  two  expressions  con- 
tains their  highest  common  divisor  (Prin.  2),  it  is  evident  that  at  the  outset 
a  simpler  expression  that  will  contain  the  highest  common  divisor  may  be 
obtained  by  adding  the  given  expressions,  giving  x^  _  6  at  +  9. 


122  ACADEMIC  ALGEBRA 

Find  the  H.  C.  D.  of 

6.  x'-\-2x-24.  Sind  2x'  +  7x-30. 

7.  2a^-aj-21  and  4a;2  +  4a:-63. 

8.  Sar^-flOx-S  and  6a;2-7a;  +  2. 

9.  2a^-6x^-\-7x-6  Siiid  2a^-^4:7^-Sx  +  9. 
10.   a^  +  9a;2_|_26a;  +  24  and  2a^-hl4a;2-}-20a;. 

11.    Find  the  H.  C.  D.  of 
3  aa^  —  4  aa^  —  lSax-\- 14:  a   and   3  aboi^  +  5  a5ie^  —  10  abx  —  42  a6, 

FIRST    PROCESS 

3  ax^  —  4  aa^  —  13  aaj  +  14  a)Sabx^  +  5  a&x^  —  10  abx  —  42  a6 

Reserve  the  common  factor  a  as  a  factor  of  the  H.  C.  D. 

Sa^-4:X^-13x-\-U)3bx'  +  5bx^-10bx-^2b(b 
3bx^-4.bx'-13bx+Ub 
6)96a^+  3bx~o6b 
9a^  +  3x  -56)3ar^-  4a;2-13a;+14 

3 

9a^-12aj2-39a:4-42(.7^ 
9fl^+  3ar^-56a; 
-15a;2_|_;^7^_^42 
-15ar^+17a;+42)9a;2+  3  a;-  56 

_5 

45a;2+15a;-280(-3 
45^^-51^-126 

~22)66a;-154 

3a;-     7)-15a;2_,_i7  3._^42(_5a,_(5 

-15x^+35  a; 

-18a;+42 
.-.  H.C.D.  =(*(3i»-7).  -18a;+42 

Since  the  arrangement  of  the  dividend,  divisor,  and  quotient 
may  be  either:  Divisor  )  Dividend  (Quotient ;  or  Quotient)  Divi- 
dend (  Divisor ;  by  using  these  two  arrangements  alternately,  the 
above  process  may  be  more  compactly  written  as  follows : 


HIGHEST  COMMON  DIVISOR 


123 


SECOND    PROCESS 


3  aa^  -  4  ax-  -  13  ax  +  14  a)3  aho^  +  5  ab^?  -  10  afta;  -  42  a& 
Reserve  the  common  factor  a  as  a  factor  of  the  H.C.D. 


6x 
6 


3a;3_    4a^-13a;  +  14 
3 

9  a^  -  12  aj2  -  39  X  +  42 
9  0^3+    3a^-56a; 


15  a.-2  +  17  a;  +  42 


-  15  0^2  +  35  a; 


-  18  X  +  42 

-  18  x  4-  42 


^hx^-^-ohx"-  10  hx  -  42  6  I 
3  &a!^  -  4  6a.-^  -  13  6a;  +  14  &  I  6 
&)9  6a;^+    3  5a;  -  56  6 

9a;'+    3  a; 

5 


-m 


45a;2_|_i5^    _  280 
^0  2(^-hlx    -126 


22)66 


bx 


154 


3a; 


H.C.D.  =  a  (3  a; -7). 


-3 


Suggestion.  —  When  the  quotient  consists  of  more  than  one  term,  for 
convenience  each  term  is  placed  opposite  the  corresponding  part  of  the  divi- 
dend or  product. 

Rule.  —  Divide  one  expression  by  the  other,  and  if  there  is  a 
remainder,  divide  the  divisor  by  it;  then  divide  the  preceding  divisor 
by  the  last  remainder,  and  so  on,  until  there  is  no  remainder.  The 
last  divisor  will  be  the  highest  common  divisor. 

If  any  remainder  does  not  contain  the  letter  of  arrangement,  the 
exjyressions  have  no  common  divisor  in  that  letter. 

If  more  than  two  expressions  are  given,  find  the  highest  common 
divisor  of  any  two,  then  of  this  divisor  and  another,  and  so  on.  Tlie 
last  divisor  will  be  the  highest  common  divisor. 

1.  If  either  expression  contains  a  monomial  factor  not  found  in  the  other, 
it  should  be  rejected  before  beginning  the  process. 

2.  A  common  factor  of  the  expressions  should  be  removed  before  begin- 
ning the  division,  but  it  must  appear  as  a  factor  of  the  highest  common  divisor. 

3.  When  necessary,  to  avoid  fractional  quotients,  any  dividend  or  divisor 
may  be  multiplied  or  divided  by  any  number  not  a  factor  of  the  other. 

4.  The  highest  common  divisor  has  an  ambiguous  sign.  For,  if  a  positive 
divisor  is  contained  in  a  dividend,  the  same  negative  divisor  also  will  be  con- 
tained in  that  dividend,  but  the  signs  of  the  quotient  will  be  changed.  It  is 
not  customary  to  write  both  divisors. 


124  ACADEMIC  ALGEBRA 

149.  The  principle,  that  the  exact  divisor  reached  by  the  process 
given  in  the  rule  is  the  highest  common  divisor,  may  be  proved  as 
follows : 

Let  A  and  B  represent  any  two  polynomials  freed  of  monomial  factors, 
the  degree  of  B  being  not  higher  than  that  of  A. 

Divide  A  by  B,  and  let  the  quotient  be  m  and  the  remainder  D  ;  divide  B 
by  JD,  and  let  the  quotient  be  n  and  the  remainder  E ;  divide  D  by  E,  and 
let  the  quotient  be  r  and  the  remainder  zero ;  that  is,  let  E  be  an  exact 
divisor  of  Z>. 

It  is  to  be  proved  that  E  is  the  highest  common  divisor  of  A  and  B. 

PROCESS 

B)A(m 
mB 
D)B{n 
nP 
E)D(r 
rE 
0 

Since  the  minuend  is  equal  to  the  subtrahend  plus  the  remainder, 
A  =  mB  +  D,  and  A  -  mB  =  D  ; 
B=  nD-\-E,  and  B-  nD  =  E;  and  D  =  rE. 

Since  the  division  has  terminated,  ^  is  a  common  divisor  of  D  and  nD 
(Prin.  1)  ;  also  of  D  and  nD-\-E,  or  B  (Prin.  2)  ;  also  of  B  and  mB  (Prin.  1) ; 
also  of  B  and  mB  +  2>,  or  J.  (Prin.  2).  That  is,  ^  is  a  common  divisor  of  B 
and  A. 

Every  common  divisor  of  A  and  ^  is  a  divisor  of  mB  (Prin.  1)  ;  and  of 
A  —  toJ5,  or  D  (Prin.  2).  Therefore,  every  common  divisor  of  A  and  ^  is  a 
divisor  of  nD  (Prin.  1)  ;  and  oi  B  -  nD,  or  E  (Prin.  2). 

But,  since  no  divisor  of  E  can  be  of  higher  degree  than  E  itself,  E  is  the 
highest  common  divisor  of  A  and  B. 

150.  The  principle,  that  the  highest  common  divisor  of  several 
expressions  may  be  obtained  by  finding  the  highest  common 
divisor  of  two  of  them,  then  of  this  result  and  a  third  expression, 
and  so  on,  may  be  proved  as  follows : 

Let  P  be  the  highest  common  divisor  of  A  and  J5,  and  Q  the  highest  com- 
mon divisor  of  P  and  a  third  expression  C. 

Then,  since  P  contains  all  the  common  factors  of  A  and  P,  and  Q  con- 
tains of  these  particular  factors  only  such  as  are  factors  of  C  also,  Q  is  the 
highest  common  divisor  of  A^  P,  and  C. 

This  method  may  be  extended  to  embrace  any  number  of  expressions. 


HIGHEST  COMMON  DIVISOR 


125 


Find  the  H.  C.  D.  of 

12.  2a^-7a^-f  2aj  +  3  and  2a^  +  7ic2_53._4 

13.  ^a^  +  l^o^-x-lO  Qjidi  Si>?  +  13x'  +  2x-S. 

14.  l-2a;-5a^  +  6a^  and  l  +  5a;  +  2i»2_8a^. 

15.  l-4a;  +  a^ H-6a;3  and  l  +  3a;-6a^-8a^. 

16.  l-a;-14a^4-24a;3  and  36a^-24iB2  +  a;  +  l. 

17.  ??z^  —  4 m^  —  20 m  +  48  and  m^  —  m^  —  14 m  +  24. 

18.  3a3  +  20a2-a-2  and  3a3  +  17a2  +  21a-9. 

19.  8  aa^  +  22  cticH- 15  a  and  6  6ar^  + 11  6x  +  3  6. 

20.  20  ft^c  -  2  6c  -  4  c  and  8  a^d^c  _  4  a^jc  +  a^c. 

21.  21ax  —  nax^  —  6a^  +  ax'^  and  7  ao;  +  34  aa^  —  5  oaj^. 

22.  a.'3-7x  +  6,  x^-23i?-9x^  +  l^x,  a^  +  ic2-4a;-4 

23.  a^-5a;4-4,  ic*-2a;3  4-l,  s^ -\-4.y? -^x-2. 

24.  l+4iB2  4-5ar',  2  +  5a;  +  3a^,  a^-4a;^  +  5a,'2-2. 

25.  34-ir-8a:2_f_4a^^  3  _8a;-8ar'  +  8ar^  16 a^ - 48 a^  +  81. 

26.  a:«-6ar^-5a^-14,  a^- 10a^4- 20a:  + 7,  a;^  -  310  a;  -  231. 

27.  Find  the  H. CD.  of  a^+a;^4-a^-a;-2  and  27^+o^-a?-x^-l. 


PROCESS    BY    DETACHED    COEFFICIENTS 


1+0+1+1+0^1 


l_2-3-l+2+3 

2+4+2-2-4-2 

2-4-6-2+4+6 

8)8  +  8  +  0-8-8 

1+1+0-1-1 


H.C.D.^ajHa^'-aj-l. 


2+1+0-1-1+0-1 

2+0+2+2+0-2-4 
1-2-3-1+2+3 


1+1+0_1_1 
-3-3+0+3+3 
-3-3+0+3+3 


-3 


Find  the  H.  C.  D.  of 

28.  ar^-ir^-2a.-3-a^  +  .T  +  2  and  ar' +  3a;^  +  3a^  +  a^  -  x  -  1. 

29.  a^  +  a;^  —  3^2  _  -^  ^  _  4  ^^^  2a^  +  3 x*  +  3 a^  +  3  a;-  —  7  i>;  -  4. 


126  ACADEMIC  ALGEBRA 

30.  a^-2x'-2o(^-lla^-x-15  and  2x-5-7a;^+4a^-15a^+a;-10. 

31.  a^-3a*-3a^-3a^-19a-15anda'  +  3a*-3a^-h9a^-a-15. 

32.  5a*4-a^-lla^+9a2-8a+4  and  2a^-a'-5a^+8a^-4:a. 
38.  x^  —  5x  +  4:  and  a?^  —  a;^  —  3 a;^  —  5 aj  — 12. 

34.  a3  +  3a2-2a-6  and  a^  +  4a^  +  4a'^  4- 4a2  -  a  -  12. 

35.  1  —  4 a^  +  3 a*  and  1  +  a  —  a^  —  5  a^  +  4 a^ 

36.  2  —  a  H-  3  a^  +  5  a^  —  a*  and  4  —  4  a  +  a^  _  g  a^ 

37.  ?/^  +  132/'^  +  202/-14  and  7  -  3y  -  20y^ +  2f  -  f. 

38.  6a^-lla^-35a;,  30 ar^ -  115 a;  +  35,  23^-5x^-5x^7. 


3j<>4C 


LOWEST   COMMON   MULTIPLE 


151.  1.  What  number  exactly  contains  2,  5,  a,  and  h,  or  is  a 
multiple  of  2,  5,  a,  and  b? 

2.  What  different  prime  factors  must  enter  into  every  number 
that  will  contain  4  a^b,  a^b^,  and  10  ab^,  or  must  be  found  in 
every  common  multiple  of  Aa^b,  a^b^,  and  10  a6^? 

3.  What  is  the  lowest  power  of  a  that  common  multiples 
of  4  a%  a^b^,  and  10  ab^  can  contain  ?  What  is  the  lowest  power 
of  6  ?   of  2  ?   of  5  ? 

What,  then,  is  the  lowest  common  multiple  of  4  a^b,  a^b^,  and 
10  ab'? 

To  what  is  the  lowest  common  multiple  of  two  or  more  expres- 
sions equal  ? 

152.  An  expression  that  exactly  contains  each  of  two  or  more 
given  expressions  is  called  a  Common  Multiple  of  them. 

6abx  is  a  common  multiple  of  a,  3  6,  2x,  and  6  abx.  These  numbers 
may  have  other  common  multiples,  as  12  abx,  6  a-b%  18  a"6x-,  etc. 


LOWEST  COMMON  MULTIPLE  127 

i53.  The  expression  of  lowest  degree  that  will  exactly  contain 
each  of  two  or  more  given  expressions  is  called  their  Lowest 
Common  Multiple. 

6  abx  is  the  lowest  common  multiple  of  a,  3  &,  2x,  and  6  ahx. 

The  abbreviation  L.  C.  M.  is  used  for  Lowest  Common  Multiple. 

The  lowest  common  multiple  in  algebra  corresponds  to  the  least  common 
multiple  in  arithmetic.  But,  since  letters  may  represent  any  numbers,  as, 
for  instance,  numbers  not  prime  to  each  other  or  fractions,  the  term  least 
is  not  applicable  to  algebraic  common  multiples. 

Thus,  the  algebraic  lowest  common  multiple  of  a^b'^,  ab^^  and  bx  is  a'^b'^x. 
If  a  =  4,  6  =  3,  and  x  =  2,  0^6%,  the  lowest  common  multiple  of  the  given 
expressions,  is  equal  to  864.  If,  however,  the  values  of  a,  6,  and  x  are  sub- 
stituted for  those  letters,  the  given  expressions  become  144,  108,  and  6 ;  and 
their  least  common  multiple  is  432. 

It  is  thus  seen  that  the  lowest  common  multiple  of  two  or  more  expressions 
is  not  necessarily  their  least  common  multiple. 

154.  Principle.  —  The  lowest  common  multiple  of  two  or  more 
algebraic  expressions  is  the  product  of  all  their  different  prime  fac- 
tors, using  each  factor  the  greatest  number  of  times  it  occurs  in  any 
of  the  expressions. 

155.  To  find  the  lowest  common  multiple  of  expressions  that  may 
be  factored  readily  by  inspection. 

Examples 
1.   What  is  the  L.  C.  M.  of  12  x'yi^,  6  a^xif,  and  8  axyz^  ? 

PROCESS 

12ar^2/z*   =2 -2  -  S  -  x^  •  y  -  si* 
6a^xy^  =2'S-a:'-x-y^ 
8  axyz^  =2-2-2'a-X'y'Z^ 


h.C.M.  =  2'2'2'3'a^'x''y''Z*  =  24:a^a^fs^ 

Explanation.  —The  lowest  common  multiple  of  the  numerical  coefficients 
is  found  as  in  arithmetic.     It  is  24. 

The  literal  factors  of  the  lowest  common  multiple  are  each  letter  with  the 
highest  exponent  it  has  in  any  of  the  given  expressions  (Prin.).  They  are. 
therefore,  a'^,  x^,  y'^,  and  z^. 

The  product  of  the  numerical  and  literal  factors,  24  a^x^y^s*,  is  the  lowest 
common  multiple  of  the  given  expressions. 


128  ACADEMIC  ALGEBRA 

2.   What  is  the  L.  C.  M.  of  x"  -  2  xy -\- y^,  a^  -  y',  smd  a^-{-f? 

PROCESS      , 

a^  —  2xy-\-y^  =  (x-y)(x-y) 
a^-y^  =(x-y)(x-j-y) 

a?  +  f  =(x  +  y)(x'-xy  +  y^) 

L.  C .  M.  ={x~yy(x-\-y)(x'-xy  +  y^ 

=  (x-  yfiix?  4-  f) 

Rule.  —  Factor  the  expressions  as  far  as  may  be  necessary  to 
discover  their  different  prime  factors. 

Find  the  product  of  all  their  different  prime  factors,  using  each 
factor  the  greatest  number  of  times  it  occurs  in  any  of  the  given 
expressions. 

The  factors  of  the  L.  C.  M.  may  often  be  selected  without  separating  the 
expressions  into  their  prime  factors. 

Find  the  L.  C.  M.  of 

3.  a^y?y,  a'^x-if,  and  ax^y. 

4.  lOa^ftV,  5  afe^c,  and  25  6V(i3. 

5.  Ua^b\  24.(^de,  and  Sea'bH^^. 

6.  IS  a^br",  12  p^qh,  and  54:  abVq. 

7.  x"*y^j  a;'"~y,  x"*-^y*,  and  x"'+^y. 

8.  ic^  —  2/^  and  x^  -\-2xy  +  y^. 

9.  a^  — 2/2  and  a^  —  2xy-\-y\ 

10.  ay^  —  y%  oc^  -\-2xy  -{-  y^,  and  x^  —  2xy  -^  yi 

11.  a^-n^  smd  Sa^  +  6a^n  +  3a7i\ 

12.  a;^  -  1  and  a^x"  +  a^  -  5V  _  ^2 

13.  a^  +  1,  ab  —  b,  a?  -\-  a,  and  a^  ~  1. 

14.  2  X  +  y,  2  xy  —  y%  and  4:X^  —  y\ 

15.  1  -\-  X,  X  —  ix^,  1  +  Qi^,  and  x^(l  —  x). 

16.  3  +  a,  9  -  a2,  3  -  a,  and  5  a  -f- 15. 

17.  a  —  b,b  —  c,b-\-a,  and  a^  —  61 


LOWEST  COMMON  MULTIPLE  129 

18.  2  a;  +  2,  5  a;  —  5,  3  ic  —  3,  and  x^  —  1. 

19.  3x-dy,  3aj2-f-27/,  and  2x-{-6y. 

20.  166*^-1,  12b'--h3b,  206-5,  and  2b. 

21.  l-2a;2_^;^,4^  (1-^)%  and  l-j-2x-^x^. 

22.  1  -  a,  1  +  a,  1  +  a-,  1  +  a^  and  1  4-  a'. 

23.  a;?/  -  /,  a^  +  a;!/,  a^y  +  y\  and  3:^  +  2/^. 

24.  a^  — 2/^,  ar-hxy-hy^,  and  x^  —  xy. 

25.  62  _  5  5  _^  6^  52  _  7  5  _^  10,  and  6^  _  10  6  +  16. 

26.  a^+7a;-8,  a:^  _  1,  a;  +  a;2^  and  3aa^  -  6aa;  +  3a. 

27.  a^  _  a,^^  a  -  2  x,  a^  +  2  ax-,  and  a^  -  3  a^a;  +  2  aa^. 

28.  m^  —  a^,  m^  +  w?a;,  7/1^  +  mx  +  ar^,  and  (??i  +  a;)ar^. 

29.  ar^-3a;  +  2,  ar^  + 4a; +  4,  ar^  + 3a; +  2,  and  ar^  -  1. 

30.  x"  -  /,  x'  -\-2(^y^  +  y\  o^-\-fy  and  x"  +  xy -\-  f. 

31.  a;^  H-  ar^2/  +  ^'Z  -^  f  ^.nd  a;^  —  a;^^^  +  a;/  _  f. 

32.  a^  H-  4  a  -f  4,  a^  _  4^  and  a*  —  16. 

33.  a?-(b  +  c)\  6^  -  (c  +  a)\  and  c*  -  (a  +  6)^ 

34.  1  -  a  +  a^,  1  +  a  +  a^,  and  1  +  a^  -f-  a*. 

35.  a*  4-  4  and  a^  —  2  a*  +  4  a  —  4. 

36.  a«  -  6-^  and  a«  +  a^h^  +  ^^ 

37.  ««  +  y^  and  a^x-^  -  6y  +  ^y  _  6V. 

38.  a^-2  a%  +  a-'i^  -  9  6*  and  a*  +  5  a^fta  ^  9  54 

39.  a*  -  a^  +  1,  a«  +1,  a*  +  a'  +  1,  and  a"  -  1. 

40.  Find  the  lowest  common  multiple  of  a;^  +  6ar^  +  5a;  —  12 
and  ar»-8ar2  4-19.x--12. 

Solution 

x8-f6x2+    6a;-12=(x-l)(x2  +  7x+12) 
=  (x-l)(x  +  3)(x  +  4). 

a:8_8x2  +  19x-12=(x-  l)(x2-7x  +  12) 
=  (x-l)(x-3)(x-4). 
.-.  L.  C.  M.  =(x-l)(x  +  3)(x-3)(x  +  4)(x-4) 
=  (x-l)(x2-9)(x2-16). 

ACAD.    ALG.  — 9 


130  ACADEMIC  ALGEBRA 

Suggestion.  — In  solving  the  following  the  Factor  Theorem  will  be  found 
useful. 

41.  a^-6x^-\-llx-6  Bind  a^-9a^-\-26x-2i. 

42.  a^-5o(^-4.x-\-20  SLiid  a^-\-2x^-25x-50. 

43.  x^-hSx^  —  A  and  x^-{-a^  —  x  —  l. 

44.  a^-4:X^-\-5x-2  Siud  a^-Sx^  -]-21x-\S. 

45.  a^  +  5x^-{-7x-{-3  SLud  x^-7x^-5x-\-75. 

46.  x3_|_2aj2_4a;-8,  a^  -  aj^ -8a;  +  12,  a.-^  +  4x2  -  3  a;  -  18. 

47.  a;3-9a;2_^23x-15,  x^  ^  3^2  _  17  3.  _p  15^  a^+7  x" -\-7  x-15. 

48.  x3_^7aj2_^i4^_^g^  a^^_3a^_6x-8,  x^  _^  ^2  _  ^q  ^^  _^.  g. 

156.  To  find  the  lowest  common  multiple  of  expressions  that  can- 
not be  factored  readily  by  inspection. 

x2-3x+    2  =  (x-l)(x-2).  (1) 

x^-5x-^    6  =  (x-2)(x-3).  ^2) 

x2-7x4-12  =  (x-3)(x-4).  (3) 

L.  C.  M.  =  (x-l)(x-2)(x-3)(x-4). 

1.  Find  the  lowest  common  multiple  of  expressions  (1)  and  (2) 
from  their  factors ;  from  the  product  of  their  factors.  By  what 
factor  of  the  two  expressions  must  the  product  be  divided  to 
obtain  the  lowest  common  multiple? 

2.  How,  then,  may  the  lowest  common  multiple  of  two  expres- 
sions be  found  ? 

3.  Since  (x  —  1)  (x  —  2)  (x  —  3)  is  the  lowest  common  multiple 
of  the  first  two  expressions,  what  factor  of  the  third  expression 
must  the  lowest  common  multiple  of  all  the  expressions  contain  ? 

157.  Principles.  —  1.  The  lowest  common  multiple  of  two  ex- 
pressions is  equal  to  their  product  divided  by  their  highest  common 
divisor;  or,  it  is  equal  to  either  of  them  multiplied  by  the  quotient 
of  the  other  divided  by  the  highest  common  divisor. 

2.  The  lowest  common  multiple  of  several  expressions  may  be 
obtained  by  finding  the  lowest  comm'on  multiple  of  two  of  them;  then 
of  this  result  and  a  third  exjjression;  and  so  on. 


LOWEST   COMMON  MULTIPLE  131 

Proof  of  Principle  1. 

Let  F  be  the  highest  common  divisor,  or  factor,  of  A  and  B,  and  L  their 
lowest  common  multiple.  Let  F  be  contained  a  times  in  A  and  b  times  in  B, 
or  let  A  =  aF  and  let  B  =  bF. 

It  is  to  be  proved  that  L  =  AJLA  ot  A  x  —,  ot  B  x  —- 

F  F  F 

Since  F  contains  all  the  commpn  factors  of  A  and  B,  a  and  b  have  no 

common  factors  ;  consequently,  since  A  =  aF  and  B  =  bF, 

L  =  abF. 

Multiplying  by  jP,  FL  =  abFF ; 

but  Ax  B  =  aF  xbF=  abFF. 

Therefore,  Ax.  1,  FL  =  AxB,  j» 

and  L  =  ^21^,  or  .4  x  |^,  or  i^  x  ^. 

Proof  of  Principle  2. 

Let  L  be  the  lowest  common  multiple  of  A  and  B,  and  M  the  lowest  com- 
mon multiple  of  L  and  a  third  expression  C. 

It  is  to  be  proved  that  M  is  the  lowest  common  multiple  of  A,  Bj  and  C. 

Since  L  is  the  expression  of  lowest  degree  that  is  exactly  divisible  by  both 
A  and  B,  and  M  is  the  expression  of  lowest  degree  that  is  exactly  divisible 
by  both  L  and  C,  M  is  the  expression  of  lowest  degree  that  is  exactly  divis- 
ible by  A^  B,  and  C. 

Examples 

1.   Find  the  L.  CM.  of  a^  +  Gar'-f  lla;  +  6  and  a:»-4«2_,_^_^6 

PROCESS 

Prin.  1,  L. C. M.  =  (^ +  6^  + jl^ +  6)(^ -4a^  + .  +  6) 

xi.  O.  D. 

^  (a;  +  l)(a^  +  5a;  +  6)(a;  +  l)(x'-5x  +  6) 

x-\-l 
=  (a;  +  1)  (a^  +  5  a;  +  6)  (ar'  -  5  a;  4-  6) 

or  (aj  +  l)(a;  +  2)(a;  +  3)(a;-2)(a;-3) 

or  (a;  +  l)(a^-4)(a:«-9) 

Find  the  L.  C.  M.  of 

2.  4  a3  +  7  a2  +  10  a  -  3  and  4  ci^  +  9  a2  +  14  a  +  3. 

3.  2a3_lla2  +  18a-14  and  2a^ -{- So" -10  a -\-U. 

4.  5a^-llar^-h3x4-12  and  5a:3-19a^4-27a;-12. 


13ti  ACADEMIC  ALGEBRA 

6.  4a^-14«2  +  22a;-8  and  2a;^-3a^-ar^H-12aj. 

6.  6a3  +  3a2-15a-75  and  2  a^ -{- 11  a^  ~\- 25  a -^  25. 

7.  4a3_27a2-2a  +  15  and  2a^  -  Qa^- 28a2- 15a. 

8.  3c3-llc2-32c-16  and  3(^ -19c^ -\-Sc -^16. 

9.  4a;*-7a^  +  7«2_i;^^_^g  ^^^^  2x*  +  i(^  -  a^  -  x  -  6. 

10.  a;^-a;^-3a;4-9  and  3  aa;^  -  3  aa;^  -  18  ax^  +  45  aa;  -  27  a. 

11.  20^  +  40a;2_|_25i»4-125  and  6a;»  +  7a.'2  +  10a;  + 25. 

12.  12  m^  -  18  m^  -|-  26  ??i  -  10  and  15  m^  -  9  m^  -h  19  m  +  10. 

13.  6a3a;-5a2a;_l8aa;-8a;  and  6a^b -ISa^^b -6ab-hSb. 

14.  4:9c^-^4:a^y-5xy^-\-25f  and  4 a^ -  16 a^?/ +  25 a^i/^ - 25 ^. 

15.  10a«  +  29a2_36a  +  9  and  Sa^  +  34:a' -^9a -9. 

16.  4a;4-17a^/  +  42/'  and  2x' -  x^y  -  SxY  -  5xf -2y*. 

17.  5a;^4-8a3-27a;2_^143._-^Q  ^^^^  3x*4-4a^-17a^+14a;-10. 

18.  2a;*-9a!«  +  18a;2-18a.'  +  9  and  3  a-^-11  a^+ 17  a;^- 12  a;+6. 

19.  3a*H-13a3-19a2+12a-4  and  ia' -\-22  0^-2  a^  +  2  a-{-i, 

20.  6a^4-5«-6,  8a;2^io^._3^  10a;24-9a;-9.     . 

21.  a;^-2a:3_^a^_;^    .^4_^2_^2a;-l,  a;*-3a^4-l. 

22.  a;^-7a;2  +  9,  aj*  +  2a^  +  a^-9,  a;^-a^-6a;-9. 

23.  a;*-4a^  +  4ar^-16,  a;^-12a^  +  16,  a?^-4ar^  + 16a;  -  16. 

24.  x'-4:X^-\-4.x^-25,   x*-4.x^-{-20x-25,  x*-Ux^-\~2o. 

25.  4a;^  +  5a;2-a;-l,  6 a;*  +  a:"  +  8 a;^ -  1,  36^^-13a^4-l. 

26.  10a;^H-7a^-33a:2^26a;-10  and  2a;^+7a.'34-5a:2_4^._;^Q 

27.  16a;*  +  16a.'3-48a^-36a;  +  27     and     24  a;^  +  20  a^  -  74  a;^ 

—  45  a;  +  45. 

28.  10a;^4-7a^H-2a^-a;-2  and  6a;3  +  5a;2  +  4a;  +  1. 

29.  5a;^-f  3ar^  +  6ar^  +  a;  +  3  and  15 a^ -j- U x' -\- x  +  12. 

30.  2ar^-a;2-3a;  +  2,  4:^^  +  60^  -  2x- 4,  4 a^  -  5 x -{- 2. 

31.  a;^-l,  2a:3^2a;2-5a;4-l,  a;'-3a;  +  2. 


FRACTIONS 


158.  A  fraction  is  expressed  by  two  numbers,  one  called  the 
numerator,  written  above  a  line,  and  the  other  the  denominator, 
written  below  the  line.     Thus,  -  is  a  fraction. 

0 

If  a  and  h  represent  positive  integers,  as  3  and  4,  the  fraction 

-  is  equal  to  - ;  that  is,  it  represents  3  of  the  4  equal  parts  of 

anything.     This  is  the  arithmetical  notion  of  a  fraction. 

But,  since  a  and  b  may  represent  any  numbers,  positive  or 

negative,  integral  or  fractional,  rational  or  irrational,  -  may  repre- 

4  ^ 

sent  an  expression  like  — .    Since  a  thing  cannot  be  divided  into 

5|  equal  parts,  algebraic  fractions  are  not  accurately  described  by 
the  definition  commonly  given  in  arithmetic.  But,  since  an  ex- 
pression like  \Q,  regarded  as  20  fourths,  is  equivalent  to  5,  or 
20  H-  4,  it  is  evident  that  the  numerator  of  a  fraction  may  be  re- 
garded as  a  dividend,  and  the  denominator  as  its  divisor ;  and  this 
interpretation  of  a  fraction  is  broad  enough  to  include  the  fraction 

-  when  a  and  h  represent  any  numbers  whatever.     Hence, 

Tlie  expression  of  an  unexecuted  division,  in  ivhich  the  dividend 

is  the  numerator  and  the  divisor  the  denominator,  is  an  Algebraic 

Fraction. 

The  fraction  -  is  read,  '  a  divided  hy  ?).' 
h 

-  159.  The  numerator  and  denominator  of  a  fraction  are  called 
its  Terms. 

y^  160.  An  expression,  some  of  whose  terms  are  integral  and 
some  fractional,  is  called  a  Mixed  Number,  or  a  Mixed  Expression. 

a  —  ^  ~    ,   - —  2  -f  — ,  and  a  —  b  -\ are  mixed  expressions. 

c        a^  x'^  ab 

133 


134  ACADEMIC   ALGEBRA 


REDUCTION  OP  FRACTIONS 

^   161.    The  process  of  changing  the  form  of  an  expression  with- 
out changing  its  value  is  called  Reduction. 

162.    To  reduce  fractions  to  higher  or  lower  terms. 

-^  163.    A  fraction  is  in  its  Lowest  Terms  when  its  terms  have  no 
common  divisor. 

1  X  3  X 

164.    1.    How   many  eighths   are   there   in  -?   in  -?   in  — ? 

m  ?   m  —  ?   m  —  ?   m  — ^  ? 

4  16  24  32 


2.  How  many  tenths  are  there  in  -?   in  —  ?   in  —  ? 

^  o  Zi) 

3.  If  a  dividend  is  multiplied  by  any  number,  as  2,  and  the 
*   divisor  is  multiplied  by  the  same  number,  how  is  the  quotient 

affected  ? 

4.  If  a  dividend  is  divided  by  any  number,  as  2,  and  the  divisor 
is  divided  by  the  same  number,  how  is  the  quotient  affected? 

5.  Since  a  fraction  may  be  regarded  as  an  indicated  division, 
what  may  be  done  to  the  terms  of  a  fraction  without  changing 
the  value  of  the  fraction  ? 

y^   165.    Principle.  —  Multiplying  or  dividing  both  teryas  of  a  frac- 
tion by  the  same  number  does  7iot  change  the  value  of  the  fraction. 

The  proof  of  the  principle  is  as  follows : 

Let  a  and  b  be  any  two  numbers,  a  the  dividend,  b  the  divisor,  and  - 
the  quotient.     Also,  let  m  be  any  number. 

It  is  to  be  proved  that  ^  =  ^. 
b      mb 

Since  the  quotient  multiplied  by  the  divisor  equals  the  dividend, 

^xb  =  a.  (1; 

b 

Multiplying  (1)  by  w,  Ax.  4,  ^  x  mb  =  ma.  (2) 

0 

Dividing  (2)  by  mb,  Ax.  5,  q^^m,  (3-) 

0      mo 


FRACTIONS  135 

Hence,  the  terms  of  any  fraction,  as  -,  may  be  multiplied  by  any  num- 

h 

ber,  or  the  terms  of  any  fraction,  as  — ,  may  be  divided  by  any  number, 

mh 

without  changing  the  value  of  the  fraction. 

Examples 

to  a  f  r£ 
a  +  6 


1.    Reduce to  a  fraction  whose  denominator  is  a^  —  6^. 


PROCESS 

(a2  -  &2)  _j_  (a  +  6)  =  a  _  6 


"    a  +  b      (a +  6)  (a -6)      a^-b^ 

Explanation.  —  Since  the  required  denominator  is  (a  —  b)  times  the 
given  denominator,  both  terms  of  the  fraction  must  be  multiplied  by  (a  —  6) 
(Prin.). 

2.  Reduce — ^  to  its  lowest  terms. 

Explanation.  —  Since  a  fraction  is  in  its  lowest 

PROCESS  terms  when  its  terms  have  no  common  divisor,  the  given 

^^     „  „         _  fraction  may  be  reduced  to  its  lowest  terms  by  remov- 

'■ — ^  =  ^       ing  in  succession  all  common  divisors  of  its  numerator 

30  a  xz       10  az       ^^^  denominator  (Prin.),  as,  3,  a,  a,  and  x  ;  or  by  divid- 
ing the  terms  by  their  highest  common  divisor,  3  d^x. 

3.  Change  —  to  a  fraction  whose  denominator  is  4  W, 

Z  0 

4     Change  —  to  a  fraction  whose  denominator  is  42. 

5.  Change to  a  fraction  whose  denominator  is  556. 

11  & 

6.  Change  — -^  to  a  fraction  whose  denominator  is  84  xy. 

14  a; 

A      2 

7.  Change  —^  to  a  fraction  whose  denominator  is  20^. 

52/ 


136  ACADEMIC  ALGEBRA 

8.   Change  ^'"~      to  a  fraction  whose  denominator  is  (x  —  If 
x  —  1 

9„   Change     ^  ~ —  to  a  fraction  whose  denominator  is  (2  ic-f  5)^ 
2a;  4-5 

10.  Change  — - —  to  a  fraction  whose  numerator  is  3  a  -f  a\ 

11.  Keduce  ^~     to  a  fraction  whose  denominator  is  a^  —  b\ 

12.  Reduce    ^~^    to  a  fraction  whose  numerator  is  ar^  —  yl 

2a;-)- 2/ 

13.  Keduce  to  a  fraction  whose  denominator  is  a  —  b. 

b  —  a 

14.  Reduce  -^ —  to  a  fraction  whose  denominator  is  4  —  ar^, 

x  —  2 

Reduce  the  following  to  their  lowest  terms : 


15 

aV 

a^xy 

16. 

17. 

a^dV 

18. 


19. 


20. 


Wxy^ 

16  m^nQt?z^^ 
40  amhj^ 

750  aft^c* 

35a^6cd« 
42  aft^cc?*' 

77  aV5^y 


22. 

-lOOa^y 

9,3 

r^+Y 

rjlflyA 

24. 

a;"* 

25. 

26. 

^m-n+J 

ax 

27c 

a'by 

Sa^b 

28. 

^m+2ry2r 

^^°    121a'b'c'  ""     2a'y^ 


FRACTIONS  137 

„     aaj-^-^+i  ^^  a^ -11  a +  24: 

Zu»     — •  45. • 

n(n-2)ab'  '  a^-\-2x^-35x 

31  ^'-^'  47  7a;-2a^-3 
a2  +  2a5  +  6='  '  2x'-\-7x-4. 

32  «'  -  2  ^^  +  ^'  48  «(a4-2  6)^ 

a'-b'  '  h(a?-4.hy 

4  g^  -  9  ar^  49  a'  +  2a^6  +  a6^ 

8a3  +  27ar''  '  a**  -  2  a^ft^  _^  a6** 

„e     Soe^V  —  Soey  _-  {c*H-5ar^  — 6a; 

oo«    ^^ ^'  oi.»  — • 

x^y-^xy  2  a;2  _  2 

3g^    3a^5-3  6«  g2.  ^-'^^  +  ^ 


33. 


37. 
38. 
39. 
40. 


.  5;5.     : . 

2a?h-2h''  x^-lOar^  +  g 

4.a'-ab^  a;«-21a;  +  20 

8  a* +  068*  '    a;^  -  26  ar^  +  25* 

2^f-'^if  g^     a;3^3^^3a;4-l. 
4  a^^y  —  32y*  a;^  +  ar^  —  4a;  —  4 

•  oo.  » 

3a«6  +  3  6'  3a'^6-3a62 

10  nx  +  10  rt?/  gg      3  ft-  +  4  aa;  -  4  ar^ 
25  ?iar^ -  25  ny^'  '   9  a^  -  12  ax  +  4:0^' 

a;"+^  —  a;"  _^     2  ax  —  ay  —  4:bx -^2  by 
^.n+s  _  ^n  '   4:  ax  — 2  ay  —2  bx-]- by 

,„       ft^+^-aV  ^o  9.a'8-13ft2a;-4ft3 

42,    ii — .  00. 

ft**"^^  -f  ft"+y  3  6a;  +  3  a;?/  —  4  ft6  —  4  ft?/ 

a^  _  y  m  —  ?;i7i  -f  ?i^  —  '}i 

44.    ^y  —  ^f  +  f.  go^    am  —  an  —  m-^  n 

x^  +  y^  '   am  —  aii  -\-m  —  n 


138  ACADEMIC  ALGEBRA 

61.   Eeduce  -— ^ — cot —    ho     *^  ^*s  lowest  terms. 
3  a^  —  8  ar  —  7  ic  +  12 

Solution 

The  process  of  finding  the  H.  C.  D.  of  the  terms  of  the  fraction  can  be 
shortened  in  some  instances  by  finding  the  sum  or  the  difference  of  the 
terms,  since  the  result  will  either  be  the  H.C.D.  or  some  multiple  of  it,  §  148. 

3  a;3  -  16  a;2 -f  25  a;  -  12 

3  x3  -  8  x^  -  7  X  +  12  ' 

Subtracting  the  numerator  from  the  denominator, 

8  a;2  -  32  X  +  24 
=  8(x2-4x  +  3). 

By  trial,  x^  -  4  x  +  3  is  found  to  be  the  H.  C.  D, 

Dividing  the  terms  of  the  fraction  by  x^  —  4  x  +  3,   the  fraction  in  its 

3x-4 

lowest  terms  is  r — -— • 

3x  +  4 

.  Eeduce  the  following  to  their  lowest  terms ; 


62. 


63. 


64. 


65. 


66. 


67. 


68. 


69. 


x^  +  3x^-25x-75 

a^-\-2x''-23x-60 
i»3-lla;2-10a;  +  200* 

4a^  +  7a;-  +  10x-3_ 
4a^  +  9.^2^14a;  +  3' 

x^-\-3x'-\-4:X-\-2 

3a^_7a;2  +  4 
5a^-17x^-j-16x-^' 

5o^-Ua^  +  22x-{-5 
5a^-18a^-\-34:X-W' 

Qi?-&x^y  +  2xy^-\-3f 
a^-\-6x'y-2xy''-5f 

g^  +  6^  +  2  c^  +  2  a&  4-  3  ac  +  3  &c 
a'-{-b''  +  c'-}-2ab-\-2ac  +  2bc' 


70. 
71. 

72. 
73. 


FRACTIONS  139 

a2  +  6^  +  c^ 4-  2 a6  -  2 ac  -  2hc 

a?  -\-W  +  c"  -  2  ah  -2  ac  +  2hc 


a2  +  62  +  5c2_2a6-6ac  +  66c 

4  g^  +  9  6^  +  16  c-  +  12  a6  +  16  ac  4-  24  6c 
4a2-962  +  i6c2  +  16ac 


166.  Signs  in  fractions. 

167.  The  sign  written  before  the  dividing  line  of  a  fraction  is 
called  the  Sign  of  the  Fraction.         • 

It  belongs  to  the  fraction  as  a  whole,  and  not  to  either  the 
numerator  or  the  denominator. 

In  —  —  the  sign  of  the  fraction  is  — ,  while  the  signs  of  x  and  3  «  are  + . 
3^ 

168.  An  expression  like  ^^  indicates  a  process  in  division,  in 

—  h 
which  the  quotient  is  to  be  found  by  dividing  a  by  6  and  prefixing 
the  sign  according  to  the  law  of  signs  in  division ;  that  is, 

—  a       .  a  -\-a       .  a 

~b^^b'  Tb-^V 

—  a  _  _  a  -f  a  _  _  a. 
-\-b~     b'                        -b~     b 

By  comparing  the  above  fractions  and  their  values  the  following 
principles  may  be  deduced : 

169.  Principles.  —  1.  The  signs  of  both  the  numerator  and  the 
denominator  of  a  fraction  may  be  changed  without  changing  the  sign 
of  the  fraction. 

2.  The  sign  of  either  the  numerator  or  the  denominator  of  a  frac- 
tion may  be  changed,  provided  the  sign  of  the  fraction  is  changed. 

When  either  term  of  a  fraction  is  a  polynomial,  its  sign  is  changed  by 
changing  the  sign  of  each  of  its  terms.  Thus,  the  sign  of  a  —  6  is  changed 
by  writing  it  —  a  +  6,  or  &  —  a. 


140  ACADEMIC  ALGEBRA 

Examples 
Reduce  to  fractions  having  positive  numbers  in  both  terms : 

a—h  _  —  2  — m 


1. 

-3 
-4 

3. 

—  a  —  x 

2x 

2. 

2 
-5 

4. 

-4c 
-h-y 

c-\-  d  2  -\-n 

6. =:!-.      8.   -^(^  +  ^). 

-a-y  5{-x-y) 

170.  Since,  from  the  laws  of  signs,  changing  the  signs  of  an 
even  number  of  factors  does  not  change  the  sign  of  the  product,  it 
follows  that : 

Principle  1.  —  The  signs  of  an  even  number  of  factors  of  the 
numerator  or  of  the  denominaipr  may  be  changed  without  changing 
the  sign  of  the  fraction. 

Since,  from  the  laws  of  signs,  changing  the  signs  of  an  odd 
number  of  factors  changes  the  sign  of  the  product,  it  follows  from 
Prin.  2,  §  169,  that : 

Principle  2.  —  The  signs  of  an  odd  number  of  factors  of  the 
numerator  or  of  the  denominator  may  be  changed,  provided  the  sign 
of  the  fraction  is  changed. 


Examples 

1. 

Show  that 

-b          b 
b  —  a     a  —  b 

2. 

Show  that 

—  a                 a 

b  —  a-\-  c     a—b  —  c 

3. 

Show  that 

2                      2 

a(b~  a)          a  (a  —  b) 

4. 

Show  that 

1                                 1 

(a-b)(c-b)          (a-b)(b-c) 

5. 

Show  that 

m  —  71                    m  —  71 

{a-c)(b-a)      {c-a)(a-b) 

6. 

Show  that 

1                                          1 

(6  -a){c-  b)  (a  -c)         (a  -  b)  (b  -c)(c- 

-a) 

7. 

Show  that 

n  —  m                                m  —  n 

_. 

(y  -x){z-  y)  {x  -z)      {x  -  y)  (y  -  z)  (z  -  x) 


FRACTIONS  141 

171.   To  reduce  a  fraction  to  an  integral  or  a  mixed  expression. 

1.  How  many  units  are  there  in  -2^?   in  ^^-  ?   in  y^  ? 

2.  How  many  units  are  there  in  i^5-±A^?    j^lQ^-^^? 

4  5 

Examples 

1.   Reduce  — -I—  to  a  mixed  number, 
a; 

PROCESS  Explanation.  —  Since,  §  158,  a  fraction  may  be  re- 

,  ,      garded  as  an  expression  of  unexecuted  division,  by  per- 

=  a  H —     forming  the  division   indicated  the  fraction  is  changed 

^  ^     into  the  form  of  a  mixed  number. 

When  the  degree  of  the  numerator  is  lower  than  the  degree  of  the  denomi- 
nator, the  fraction  cannot  be  reduced  to  an  integral  or  mixed  expression. 

Reduce  the  following  to  integral  or  mixed  expressions : 

A- •  1/6.       


a;  +  2 
-     4:0i?-%x'  +  2x-l  ,,     a8-f-9a2  +  24a-f-2^ 

o,  „ •  Xo.     — • 

2x  a-f  3 

^    ah-bc-cd-^  d\  ^^    ^-^  _  6ar^  + 14a;-9 

h  '  '  x-2 

0^3?  —  aoi?  —  x  —  1  ,_     x^  —  Za?-\-6x  —  l 

5. 15. 

ax  cc  —  3 

a^-a;-15  _     g^^  3^252^54 

o. •  xo.  » 

a;  -  4  a^  +  lx" 

4g^ -I- 22a; +  21 
2a;  +  4 

/c3-3a;2-t-4a;-3 
a;  —  4 

g^-f  2a&  4-6' H-c' 
a  4-  6  -f  c 

a3-6a26  4-12a6'^-10  63 


7     ^-2a;2/-y^ 
x-y 

17. 

20^ 

18. 

^     a'-2a-26_ 
a->r^ 

19. 

0     a3  +  2a6-h6^ 

20. 

a  +  6  ct  —  2  6 

11         a^-hy*  21     ^^  +  4 a^^y  +  6  a^/  +  4  a;?/^^ 

'   x'-xy  +  Z  '  x  +  y 


142  ACADEMIC  ALGEBRA 

172.   To  reduce  an  integral  or  a  mixed  expression  to  a  fraction. 

1.  How  many  fifths  are  there  in  6  ?  in  10  ?  in  a  ?  in  3  6  ? 

2,  How  many  fifths  are  there  in  6i  ?  in  a  +  —  ?   in  2  a  +  -  ? 

5  5 

Examples 

1.   Reduce  a  +  -  to  a  fractional  form, 
c 

PROCESS  Explanation.  —  Since  1  =  c-^  c,  a  =  ac-^c. 

Since,  §158, 


a  =  — 

c 


ac  b  ,  b 


means  6  -=-  c,  a  +  -  =  ac  -^  c  -\-  b  -i-  c. 


b  _ac      b  _ac-\-b     §  104,  3,  =  (ac  +  &)  ^  c. 

Rule.  —  Multiply  the  integral  part  by  the  denominator  of  the 
fraction;  to  this  product  add  the  numerator  when  the  sign  of  the 
fraction  is  plus,  subtract  it  when  the  sign  of  the  fraction  is  minus, 
and  ivrite  the  result  over  the  denominator. 

If  the  sign  of  the.  fraction  is  — ,  the  signs  of  all  the  terms  in  the  numera- 
tor must  be  changed  when  it  is  subtracted. 


Reduce  the  following  to  fractional  forms : 

2.   a  +  t  8.    a-^^^. 

2  b 

o  V  ^  a—b—c 

4.    5c +  «'^.  10.    b-^-^^. 

6.   36_!iJl^lii;.  11.   „_c-a-6. 


c 
a +  262 

b 
1-x 

3 
2bc-l 

6.    Ax-^^^-  12.    ^±A±f_c. 


7.   4:b-\-^^^ — -'  13.    6  + 


2 

g- 4-  ab 
ab  -  b^' 


FRACTIONS  143 


1*.    a-^-x 


17.    oc^ -{- xy -j- y^ -\ — ^ 


a  —  X  ^  —  y 


16.   x  +  5-5!±i.  18.   3a-e,b-^±^^^. 

x  —  4:  a-\-2b 


16.   aJ'-ab  +  b^ ^-  19.    x-a^-a^ 

a  +  6  1-f  OJ 

173.   To  reduce  dissimilar  fractions  to  similar  fractions. 

1.  Into  what  fractions  having  the  same  denominator  may  |, 
I,  and  f  be  changed  ? 

2.  Express  — — ,  — -,  and  -— -  by  fractions  whose  common  de- 

3ic   5  a;  15  a; 

nominator  is  the  lowest  common  multiple  of  the  given  denomi- 
nators. 

-'  174.   Fractions  that  have  the  same  denominator  are  called 
Similar  Fractions. 

175.  Fractions  that  have   different    denominators   are  called 
Dissimilar  Fractions. 

176.  Principle.  —  The  loivest  common  denominator  of  two  or 
more  fractions  is  the  lowest  common  multiple  of  their  denomiriators. 

The  abbreviation  L.C.D.  is  used  instead  of  Lowest  Common  Denominator. 

Examples 

1.   Keduce  -^  and  -^  to  similar  fractions  having  their 
3  6c  6  a6  ^ 

lowest  common  denominator. 

Explanation.  —  Since  the  L.  C.  D.  of  the 
given  fractions  is  the  lowest  common  multi- 
ple of  their  denominators  (Prin.),  the  lowest 
common  multiple  of  their  denominators  must 
be  found.     This  is  6  ahc. 

To  reduce  the  fractions  to  equivalent  frac- 
tions having  the  common  denominator  6  ahc, 
§  166,  the  terms  of  each  fraction  must  be  multiplied  by  the  quotient  of  6  a 6c 
divided  by  the  denominator  of  the  fraction. 


PROCESS 

a 

a  X  2a 

2  a' 

3  6c 

3  6c  X  2  a 

6abc 

c 
6a6 

c  xc 
6a6  X  c 

6abc 

144  ACADEMIC   ALGEBRA 

Rule.  —  Find  the  lowest  common  multiple  of  the  denominators 
of  the  fractions  for  the  lowest  common  denominator. 

Divide  this  denominator  by  the  denominator  of  the  first  fraction, 
and  midtiply  the  terms  of  the  fraction  by  the  quotient. 

Proceed  in  a  similar  manner  with  the  other  fractions. 

All  mixed  expressions  should  first  be  reduced  to  the  fractional  form,  and 
all  fractions  to  their  lowest  terms. 

Reduce  to  similar  fractions  having  their  L.  C.  D. : 

2.   I  and  ^.  8.   '^^=^,  2,  -^L_. 

2  5  a  m  -]-  n        ' 

ab        ■,  X  4:  be     Sac     7  ab 

^-  T  ^°'^f  ®-  3^'  n\'  6^" 


a 


4.x  _     3a6      7a2 


•    3  ^^     ey'  '    8aV    4b''6    a'bc' 

^     2a  ^.  Sx 

5.    — -  and  - — 

5b  4a 

„     a^b        -,   ab^ 

7..  - —  and 

2xy  4:  ay 

x" 


11. 

3-6      3 

xy'  ^f'  ^y 

12. 

x  +  y    2  X  —  y 
2    '         4      ' 

x'  -  y\ 
6 

13. 

a-2b     2b-a 

...  J              , 

X             y 

«-r. 

14. 
15. 
16. 
17. 
18. 


x^  —  \     x-\-\     x  —  \ 
a?  a  2  a 


i4_16'    a2  +  4     a^-4 
4a         36  1 


a  —  b     b  -\-  a     o?  —  b^ 
a  X  —ax 


\  —  ax    \-\-  ax     ax-\-\ 

1  1  1 


x'^lx^-W    x'^x-2     Q^-\-4x-5 


iq  a4-5  ft  — 2  ft  4- 1 

•    a2-4a4-3'    ft^  -  8  a  +  is'    a^-^a^h 


FRACTIONS  145 

ADDITION  AND  SUBTRACTION  OF  FRACTIONS 


4     3 

7 

2 

a      b 

1      2 

-  +  -; 

-  +  -; 

-+- 

X        X 

a 

a 

y    y 

X     y 

2.  What  kind  of  fractions  can  be  added  or  subtracted  without 
changing  their  form  ? 

3.  What  must  be  done  to  dissimilar  fractions  before  they  can 
be  added  or  subtracted?  How  are  dissimilar  fractions  made 
similar  ? 

178.  Principle.  —  Only  similar  fractions  may  be  united  by 
addition  or  subtraction  into  one  term. 

Examples 

1.  Find  the  algebraic  sum  of  f  -|-  ^  -  ^. 

0     b     0 

PROCESS 

§104,3,  ?  +  f_^  =  ?^t|^ 

b     b      b  b 

2.  What  is  the  sum  of  ^,    ^,   and    5|? 

PROCESS 

Sx     Tjc     5y 
4       10  "^12 
'  ^45a;     42a;     25y 
60        60        60 
^S7x-\-25y 
60 

Explanation. — Since  the  fractions  are  dissimilar,  they  must  be  mada 

similar  before  they  can  be  united  into  one  term  (Prin.)-    . 

mi-    1         ^  J  .     ^     .   ^^     3  X     45  0!     7  a;     42  x     5  w     25  M 

The  lowest  common  denommator  is  60.    -^  =  — —  ;  —  =  — — ;  ~  =  — ^' 

Therefore,  the  sum  is  45.     42x     25y^45x  +  42x  +  25y^87x  +  25y, 
60        60        60  60  60 

ACAD.    ALG.  — 10 


146  FRACTIONS 

3.   Find  the  algebraic  sum  of  ^^Lzl _ 5^zi^ _|_ ^-^ 

PROCESS 

5a;-l      3a;-2      a;-5^35a;-7      (24a;-16)      14a:-70 
8  7  4  BQ  6Q  m 

^  35  a;  -  7  -  (24  a;  -  16)  +  14  a;  -  70 

^35a;- 7 -24a; +  16  + 14a;- 70 

^25a;-61 
56 

Suggestion.  —  When  a  fraction  is  preceded  by  the  sign  — ,  it  is  expedient 
for  the  beginner  to  inclose  the  numerator  in  parenthesis,  if  it  is  a  polynomial, 
as  is  shown  above. 

KuLE.  —  Reduce  the  fractions  to  similar  fractions  having  their 
lowest  common  denominator. 

Change  the  signs  of  all  the  terms  of  the  numerators  of  fractions 
preceded  by  the  sign  —,  then  find  the  algebraic  sum  of  the  numera- 
,  tors  J  and  write  it  over  the  common  denominator. 

1.  Reduce  the  resulting  fraction  to  its  lowest  terms,  if  necessary. 

2.  The  integral  and  fractional  parts  of  mixed  expressions  may  be  united 
separately. 

3.  An  integer  may  be  expressed  as  a  fraction  whose  denominator  is  1. 

Find  the  algebraic  sum  of 

4    ^  —  ^\  ^  —  ^  q     &  —  c     g  —  c 

ab  be  '      be  ac 

^     a  -\-b  ,  a  —  b       -                       i/^«  +  ^      a  —  b 
o.    — -\ •  1U« r* 

a  —  b     a-\-b  a  —  ba-\-b 

6.  -2^  +  a.  .      11.   ^^'-2. 
a  —  X  a^  —  b^ 

7.  „  +  6  +  ?!±»l.  12.    ^^y-t±f. 

a—b  '  x—y 

8     — ^ii— +  -A_.  13        ^         x-2 

a;2  +  a;+l      a;  —  1  *    x  —  2.      x-\-2 


FRACTIONS  147 

Simplify : 

14     ^^  +  1  I  a;-2 _ x  —  ^     b  —  x 

■        3  4  6  2    * 

a;_2      ir-4  ,  2-^x     2x-^l 
6  9  4  12 

X  —  1  _x  —  2  _AiX  —  Z      1  —  X 
-      '    ~3  18  2^         6 

,„     2-6a;,4.T-l      5a;-3      1  -  a; 

"•  -^-  +  -^ 6 3-        • 

a;4-3      a;  —  2      a;  —  4      a;  +  3 

•  4  5  10  6     ' 

19    ^::ii_£z^4-2-^^±^. 

*  3  8  6 

20.    — —  +  x —, 

^,     l-2a,2a-l      2a-a^-\-l 


22. 


5  4  8 

3-}-a;-a:^     1-x^x^     l-2a;-2a^ 
4  6  3  ' 


23.    ^  +  ^_3+i.  30.      a-^>    ^«^  +  ^^         « 


5       2  6  2(a+6)      a'-fc^     a—b 

•    a  +  6      a-b     a^-b^'  *    o^  _  9     ^^  +  5 '♦"^.^.a' 

25.  ^±^  +  ^Zl^  +  ^^.  32.    2a-36-i^^f 
a  —  a;a  +  a;a^  —  ar'  2aH-3f> 

.  -I    ■  a^  —  3  ««     o         o        8a^  — 4  a; 

26.  a;  +  l  + ^.  33.    3a-2a;-- — -  • 

a*  —  1  3a  +  2a; 

a^  ?7^  —  7i 

OR        g  +  l      ■      g-l  35  1 1___L,1. 

^°-    a'+a-\-l'^a'-a  +  l  '    2i,x-l)     2  (.i- +  1)  ^  ar* 

29.    3x+-^-(2x-\-^\  36.    l  +  l+T^— 2. 


ax 


148  ACADEMIC  ALGEBRA 


37.    ^ ^^-^4-      ^ 


a  — 2     a-\-2     4  —  a- 
SuGGESTiON.  —By  Prill.  1,  §  109        ^ 


4  -  a2     a2  _  4 


38.    ^±l^^^zl^    2a^ 


a  —  1      a  +  1      1  —a- 
39.   ^^+2^     2  3 


x^-4.      x-2     2-x 
^^     x{a-\-x)     ^ax-x'  ^  ^^^ 


41     «■  +  6      a^-\-h'^     h~a 
'   a-b     b^-a"     a-\-b 

42.    _^_  +  _-l^_+      1 


43. 


x{x  —  a)      a(a  —  x)      x  —  a 

-1 L_+_J_. 

a3  -f  8      8  -  a«  ^  4  -  a^ 


AA  X   —   1  ,  X 

44. (- 


'45.    ?-+-^ ^^^+       1 


a;      l-2a;      4a^-l      l  +  2a; 

46.    ^^  +  ^^  I         ^  +  «  2 


aj^  —  a^     «^  +  aa;  +  a^     a;  —  a 
47  3  m  m  +  2  a;  5 


48. 


(m  —  2  a;)2      (m  +  a;)  (m  —  2  a;)      m  +  a; 

3 2 

2/^  —  m^/  —  12  m^     y^  —  ^  ^^.V  +  4^m^ 


49^  «& 06 ^ 


50. 


51. 


1  1.1 


x^-3x-l-2  ■  a^^2x-S     a^  +  a;-6 
2  11 


a:2  +  5a;+6     x^-{-6x-{-S      x^  +  lx^l2 


52      ''^(a^-3)  2(a?  +  2)  a;  - 1 

*a^-a;-2      a;2  +  4a;  +  3      {^  -  x  -  x^' 


FRACTIONS  149 


--     a  —  b  —  c     a  — 6  +  c,  4ac 

OO. ; 1- 


a  +  6  +  c     a-\-h  —  G      (a -\- by  —  c^ 

54.    Simplify  a' -h2a -^1  _        a^ -2a -^1^ 
^     ^    a'-2a-\-l         ^a2  +  2a4-l 

Solution 

a-2_2a  +  l  a-^  +  2a  +  l      V        a2-2a  +  l/  \        a2  +  2a+lj 

_        4a  4a  I6a^ 


a2_2a+l     a2+2a  +  l      (a-l)2(a+i)2 

Suggestion.  — Frequently,  by  reducing  one  or  more  of  the  given  fractions 
to  mixed  numbers,  the  integers  cancel  each  other  and  the  numerators  are 
thus  simplified. 


Simplify : 


^^     a'-{.2ab  +  b^     ^   ,     2ab 


56. 
57. 


a^-\-b^  a'-b^ 

d'  +  3ab-\-2b^     a^-13b^ 

ar^  +  a?  +  l       a^  —  x  +  l 


gg     a^  +  a^  +  a:  +  l      ^         3 


ar*  —  ar^ -}- a;  —  1  a;  — 1 

--     a;  +  l,a;  — 1      x-\-2      x- 
oy.   —  -\- 


x  —  1      x-\-l      x  —  2     x-\-2 

on     a;H-3      «  —  3,a;  +  4      a:  —  4 
oo. 1 • 

a;  —  3      a;  +  3     a;  —  4      a;-|-4 
a  a  2ab         4a6^ 


61. 


a-b     a  +  b     a^+ft^     ^^  _^  6^ 

Solution 

Combining  first  two  fractions,  — ^ ^  =  — -^ —      (1) 

a  —  6     a  -\-  b     a^  —  b^ 

Combining  (1 )  with  the  third  fraction,      -^^,  -  -^^^,t,  =  4^*     ^^^ 
Combining  (2)  with  the  fourth  fraction,  4^  "  4^4  =  4^"     ^^^ 

Of    —  0  d    ~j~  0         d    —  0 

Hence,  «  «  ^  «^         *«^^'         ^«^' 


6      a  +  6      a2  +  &2     «*  +  6*     a^  -  ^s 


150  ACADEMIC  ALGEBRA 


_„     a  +  &     a  —  b       Aab         Sab^ 

62. — : n    .    ,o  ~r 


a-b      a  +  b      a'-j-b^     a'-^b* 
63.    -K-^-JK-,+    ''' 


a-b     a  +  b  a^-\-b^  a* +  5* 

64        1,1  ^  ^  +  ^x 

'    x  —  1      x-\-l  a^  +  l  x^  -\-l 

a  +  x     a^-\-a?  a  —  x  o?  —  o?     4  g^a; -h  4  aar"^ 

'   a  —  x     gC-  —  x^  a-^x  c^  -\-x?          a^  —  x^ 

66.    Simplify 


(6  —  c)  (c  —  a)      (a  —  c)  (a  —  b)      (b  —  a)  (c  —  b) 


Solution 
b 


(6-c)(c-a)      (a-c)(«-6)      (b-a)(c-b) 

8  170  =  «  .  &  .  ^ 

^        '  (6_c)(c-a)      (c-a)(a-6)      (a-6)(6-c) 

_  a(a-  b)+  b(b  -  c)+  c(c  -  a) 
(&  —  c)(c  —  a)(a  —  &) 

Simplify : 


(6  —  c)  (a  —  c)  (c  —  a)  (a  —  b)  (b  —  a)(b  —  c) 

68.           «  +  l  I           ^  +  1  ,           ^  +  1 

(a  —  b)  (a  —  c)  (6  —  c)  (6  —  a)  (a  —  c)  (b  —  c) 

c^ab  bha  a^bc 
69. 


70. 


(c  —  a)  (6  —  c)      (5  —  a)  (6  —  c)      (a  —  6)  (a  —  c) 

6  — c c  —  a a  +  b 

(b  —  a)  (a  —  c)      {b  —  c){a  —  b)      (a  —  c)  (6  —  c) 


c  +  g 6  +  c ^ g  +6 

(a  —  b)(b  —  c)      {c  —  a){b  —  a)      (c  —  6)  (a  —  c) 

c  +  a  c  —  b  a  —  b 

72. 


73. 


(a4-^)(&-c)      (c-a)(a4-&)      (b-c)(a-c) 

c  -[-a—b       ,       6  +  0—  g a-{-b-{-c 

(a  —b){b  —  c)      (c  —  a)  (g  —  6)      (5  —  c)  (g  —  c) 


FRACTIONS  151 


MULTIPLICATION  OF  FRACTIONS 

179.  1.    How  much  is  5  times  |?    2  x  f  ?    6x^?    3  x  ?^? 

4x^^?  3x^?  cx^?    «x^? 
5  4  4  d 

2.  Express  5  x  ^  in  its  lowest  terms ;  3  x  | ;  4  x  ^;  2  x  -^; 

11  x^-   7x— •   10  X—-   8x— . 
^^"^22'    ^"^14'   ■'''''20'   ^"^  16 

3.  In  what  two  ways,  then,  may  a  fraction  be  multiplied  by  an 
integer  ? 

4.  How  much  is  i  of  Y,  or  ¥^5?  1  of  — ,  or  — h-4? 

5.  How  much  is  ^  of  |,  or  J  --  5?  \  of  — ,  or  — -f-4? 

6.  In  what  two  ways,  then,  may  a  fraction  be  divided  by  an 
integer  ? 

180.  Principles.  —  1.  Multiplying  the  numerator  or  dividing 
the  denominator  of  a  fraction  by  any  number  multiplies  the  fraction 
by  that  number. 

2.  Dividing  the  numerator  or  multiplying  the  denominator  of  a 
fraction  by  any  number  divides  the  fraction  by  that  number. 


Examples 


1.    Multiply  I  by  ^. 
0         a 


PROCESS  Explanation.  —  To  multiply  ^  by  -  is  to  find  c  times 

h        d 
a     c__ac 

b      d      bd         *^®  ~  P^'^  ^^  "•    ~  P^^^  of  -=  —  (Prill.  2),  and  c  tmies 

Ct  0        u  0        001 

^  =  ^  (Prin.  1).     Therefore,  «  x  ^  =  ?^. 

bd     bd  •  h     d     od 


To  find  the  product  of  ^  x  -• 
0      a 

Let  *  =  ?'<?  W 

0    a 


152  ACADEMIC  ALGEBRA 

Multiplying  each  member  of  the  equation  by  &  x  d,  Ax.  4, 

-  X- 
b     d 


§  104,  1,                                   =z^xbx-xd  = 

b            d 

-ac. 

xbd  =  ac. 

- 

Dividing  by  bd,  Ax.  5,          x  =  ^. 

bd  ^ 

...  Eq.  (1)  and  (2),  Ax.  1,           -%|  =  ^.' 

Similarly,                               f^S^^^^.' 
0     d     f     bdf 

and  so  on  for  any  number  of  fractions. 

(2) 


Rule.  —  Multiply  the  numerators  together  for  the  numerator  of 
the  product,  and  the  denominators  for  the  denominator  of  the 
product. 

1.  Cancel  equal  factors  from  the  numerator  and  denominator. 

2.  Reduce  integral  and  mixed  expressions  to  the  fractional  form  before 
multiplying. 

2.    Simplify ^!— - —  X X 


a;  +  3  x  +  b      2a^  — 10  a; 


Solution.  3x^+15.^x1-9^  1 


cc  +  3  x  +  5      2a;2-10x 

^3x(x  +  5)^.^  (a;  +  3)(x-3)^^  1 

ic  +  3  ic  +  5  2ic(a;-5) 

Canceling,        =|f^^- 
2(x  —  5) 

Multiply : 

3     ^  bv  -?^-  6.    i^  by    -1^. 

4  a;?/     -^    3a2  3  a;?/     -^        16m^ 

5^2/  V      ^_ax  2ax^  ,       _10&!. 

•    2ac     ^   10/  '    1262/  a;" 

5.    t^  by  ?^.  8.    '^^  by      ^^ 


lOc^     "^     a^  4  a;  a'^-^6'" 


FRACTIONS  158 

^m+i  jm+1  •  ^2  25-lOaj 

»•    j^.  by  — .  11.   20^8^  by  — ^^. 

10.   ^^  by  -1-.  12.   ^-^^•  +  «^  by  2^-. 

Simplify  the  following : 

a  +  0       a^  —  ah       a^  —  o^ 


14. 
15. 


jc*  ^^  a-\-  X  ^a^  —  ax  -\-  7? 


«8  ^  a^     a^  -  ir2         («  +  xf 

^a  —  h  ^       2ft  ^7?  —  y^ 

2x-\-  y     4  ft^  —  a6  4 


£+2     3^-27     __4_, 
17.    i^'-g"  X  ^~^    X      ^' 

(p-qY    p^+pq    p'^-^^i^ 

W    a^±^     ft^+^a_+4 
*   ft»-8     ft2_2a  +  4 

ft*  —  aa^  (}?  —  ax-\-Q^ 


20. 


ft* +  4      ^,   ft2  +  ft-f  1 


ft*H-a2  +  l     ft2  +  2ft  +  2 


•  r^j^^x^h     a^-{-7x-\-12 

fl^-^-3a;-10      ar^-10a;  +  21 

•  ar^_4a;-21       a^4-7a;  +  10* 

a^-\-a^+ac-\-bc  ^  a^-ax-\-ay—xy  ^  a^—x(y-a)-ay 
ax—ay  —  x^+xy      a^-\-ac-\-ax+cx     a-  —  a(y—b)  —  by 

a^_5a.^  +  8a;-4        a^  -  10  a.-^  +  33  .^  -  36 
a^-8ar^  +  19a;-12        a^  -  6a;2  +  11  .t  -  6  * 

.^_3a;3_23a.^-f.75a;-50        a.-^  -  10 a.-^  +  29 a;  -  20 
^^'    ;^_5a^-21a.-2-hl25a;-100      a.-^  -  12  a^^  _^  45  a;  -  50* 


154  ACADEMIC  ALGEBRA 

DIVISION  OF  FRACTIONS 

181.  1.   How  many  times  is  \  contained  in  1  ?    i  in  1  ?    i  in  1  ? 

2.  If  the  numerator  of  a  fraction  is  1,  how  does  the  number  of 
times  the  fraction  is  contained  in  1  compare  with  the  denominator  ? 

3.  How  many  times  is  -  contained  in  1  ?     in  1  ? 

d  X  -\-  y 

2  c 

4.  How  many  times  is  |  contained  in  1  ?     -  in  1  ?     -  in  1  ? 

182.  The  reciprocal  of  a  fraction  is  the  fraction  inverted. 

For,  since  -  multiplied  by  —  produces  1,  §  102,  if  1  is  the  dividend  and 
a  b 

-  the  divisor,  -  is  the  quotient. 
b  a 


Examples   . 


1.   Divide  -  by  — 
b     ^   d 


Explanation.  —  The  fraction  -  is  contained  in  1, 
PROCESS  ^^ 

^      ^       ,j      r^ri      d  times  ;  and  -  is,  therefore,  contained  in  1,  -  part 

b      d      b      c      be        r  J  4.-  d  1- 

oi  d  times,  or  -  times. 
c 

Since  -  is  contained      times  in  1,  it  will  be  contained  -  times  -,  or  ^— 

b  c  be 


(1) 


d 

c 

times  in  — 

b 

OK 

To  find  the 

quotient 

of 

b 

c 

d 

Let 

-t 

c 
"d 

§102, 

X  X 

.  c      a 
d~b 

Multiplying 

each 

member  of  thi 

is  equation  by 

d 

c' 

xx-x 
d 

H^ 

d 
c 

That  is. 

.  =  |x 

c 

(2) 


Hence,  eq.  (1)  and  (2),  Ax.  1,  ^  -  -^  =  ^  x  -• 
b     d     b     0 


FRACTIONS  155 

KuLE.  —  Multiply  the  dividend  by  the  reciprocal  of  the  divisor. 

1.  CJiange  integral  and  mixed  expressions  to  the  fractional  form. 

2.  An  integer  may  be  expressed  as  a  fraction  by  writing  1  for  its  de- 
nominator. 

3.  When  possible,  use  cancellation. 


4.    lby«-* 


Divide : 

2.    1  by  ^. 

X 

3.    1  by  J. 

Write  the  reciprocal  of 

«-r 

8.    ™. 

n 

3m 
P 

9.    ^-. 
3m 

Simplify  : 

,-     5mn      lOm^n 

21 

5.    Iby 


a-f  6 
a-3 
a +  3* 


10.   ^-^. 
h-y 

n.  A. 
a6 


66a;   '    3aar'  "*'   ar^-5a;  +  4  '  a2-|-a;-20" 

^3     12a^^4ax  22  g^  _  ft^  gg -|- 6' 

25ac   *  24c2'  *   a2-2a6  +  62  '  a^  -  ab 

14.    ^^^-afto;.  23.    ^^±t.^t±m±t. 
7  a^  —  y^  x  —  y 

5ab  ^25^  24      g^  4-  6^      g^  -  «&  +  jf/ 
'   3aV  •  15a2*  *   a'-Ab'  '      a-2b 


a«4-6' 

a'-Ab'' 

m^-f 

wV-y' 

m^x  H-  m^ 

16     Z^_!_?L^!.  25      m^  —  y^    ,  771^  -f  m^y  +  my' 
4  y*  *     14  a  w.^y^  —  y^  '         my^  +  y^ 

7>iy  —  y^,       if     ,  26      m'^a;  +  m^   .  m V  —  mx*^ 
(m  +  2/)^  *  w?  —  y^  m^x  —  m«^       mV  4-  x^ 


156  ACADEMIC   ALGEBRA 

31. 
32. 


07^  +  20^ -19a; -20  ' 

a.'«  +  10a:2^29aj  +  20 

a^-Wx'-^- 74.x -120 

.  x'-9a^-\-26x- 

-24 

iic^  —  5oi^  —  x-^5 

'   a^-6af  +  llx- 

-6 

a2  +  62_c2  +  2a6  .  a" 

-  62  -j_  c2  -  2  ac 

t2  _  2,2  _  ^2  ^  2  6c      a2  -  62  -}-  c^  +  2  ac 


gg -I- a.-2  —  ^2  _|.  2  ga;  .  a^  —  a^  +  y^ -\-2ay 
a^  —  x^  -{-y^  —  2ay  '  a^  —  x^  —  y^  —  2xy 

34.   Simplify  (2,  -  a:  +  J)  ^(|+ 1). 


Solution 

('- 

-|>(^^) 

=  t: 

y           '     x2y2 

a;2- 

xy  +  y^  .^                x2t/2 

_    ^H 

y            '^(a;  +  y)(a;2_ 

xy  +  2^2) 

a;  +  y 


Simplify : 

3a; 


V     y^    y^J    \     y    yy 


FRACTIONS  157 


COMPLEX  FRACTIONS 

183.    A  fraction  one  or  both  of  whose  terms  contains  a  fraction 
is  called  a  Complex  Fraction. 

It  is  simply  an  expression  of  unexecuted  division 

Examples 
a 

1.    Simplify  the  expression  -• 

X 

y 

PROCESS 

a 

X     h  '  y     h     X     hx 

y 

Simplify  the  following  expressions : 

x  +  y                            2  +  — 
2     -^-                           6     _±^. 
^  a-\ 


8. 


ab' 

-^ 

a 

3m 

m 

X 

X 

X 

m 

6. 


3 


b- 

b~ 
2 

c 

-c 

ax 

2 

2  " 

-ax 

x  +  y 

x  +  y 

y 

f 

x 

1 

1 

y 

X 

1 

X 

7.    ?.  10. 

1  +  i 

11.    Simplify  the  expression  ^ — • 

^  +  '^  +  1 

y     y 

Solution. — Multiplying  the  numerator  and  denominator  of  the  fraction 
by  y'^,  the  L.  C.  1^.  of  the  fractional  parts  of  the  numerator  and  denominator, 

the  expression  becomes  ^  ~  ^V  +  y 


158  ACADEMIC  ALGEBRA 

Simplify  the  following : 

a^-1  oc^-^f  g^  +  / 

X  xy 

xA-1  XT  —  xy  -^  2/ 


Qby 


1 


13.  f  y+\       15.  -"+^ 


1-  1 


16. 

^y 

x_y 

y    ^ 

1 

l-a 

17. 

a 

x     y-\-z  a  +  1  1  +a 

a;-2+-i-  6a-l-i 

X  4  2  ^. 

18.    Z-'  20. 


x-2  3a 

l  +  i  +  i  a;  -  5      ^  ^  24 

^     X     a^     a^  ^2  a? 

19.    = ::-•  21 


14-?  +  ^  3a.^-9 

£c     ar^  a; 


aj  +  l         a;4-l      1  —  x 

22.  — z_^+_r_  +  — _. 


1  +  x      1  —  X      1  -{-  X 


X  —  1       71  —  1       z  —  1 

+  - -\ 

gg  3a^2!  X  y  z 


yz-{-zx-{-xy  1  i  ^  i  1 

X     y      z 


12  9 


x'Jria^-^x 

-\-ab 

a?- 

-  (a^b)x 

+  ab 

x^  -  b' 

-By 
-^4-     ^ 


26.    ?-^  +  f 


1  + 


&2  +  c2  -  a3 


a     6  4-  c  2  6c 


27. 


28. 


29. 


FRACTIONS  159 

a?  +  y  I  a^  +  y^ 

a;  -  2/      ^-y^  {x-  yf 


X 
X 

-2/      a^-2/«-a:^  +  a:2y2^2^ 
+  2/      a^^^/" 

a  — ft       b  —  c 
l-\-ab     1-hbc 

1-^ 
a 

1 

(a  -b)(b-c)     ' 

^^c 

(1  +  ab)  (1  +  be) 

«+•- 

/  4  . 

184.   An  expression   of  the   form    — is  called  a 

Continued  Fraction.  b  -\ 


d  + 


/+ 


30.    Simplify 


1  + 


^-\ 


Solution 

1 


1  +  — ^     1+     ^ 


1  +  1  ^±i 

X  X 

1 


1  + 


x-\-\ 
____x_-fj_ 
x  +  \  +  x 

^  a;  +  l 

2a;+  1 

Suggestion.  —  In  the  above  example,  the  part  first  simplified  is  the  last 

complex  part :j-,  wnich  is  reduced  to  a  simple  fraction. 

1  +  5 
Every  continued  fraction  may  be  simplified  by  successively  reducing  its 
last  complex  part  to  a  simple  fraction. 


160  ACADEMIC  ALGEBRA 

Simplify  the  following : 

31.    J 34.    

x-i x  —  2 


3-x  x-2 

a 


36. 


a-\-l-\ a-\- 


a+1--  a+- 

a  a 

2 
33.    jT 36.    l-\ 


1+C  + 


2c 


2-x  ^c 

REVIEW   OP   FRACTIONS 

185.    Reduce  to  their  lowest  terms : 

,^       x^-^x^-^x-3  g     a;3_^^_22a;-40 


a^  +  Sx^-^5x-\-3  a^-7x'-\-2x-{-^0 

a^_a^-a;-2  .-,    a^^  +  lOa^'  +  Ta^-lS 

'   i^^Sx'  +  Sx-\-2  ^-    a^-Sx'-llx-^-lS 

•   a^_3a;2_8^._j^Q'  ^-^  6  a;^  - 17  aj2  +  14  a;  -  3* 

Simplify : 

9.     ^     I     y  y-^ 


2y-l      22/  +  1      1-4 


10.  — ^^ (     ^      4-      1  l-« 


4(l-a)2     ^8(l-a)      8(a  +  l)      4(( 

11     2a;-l      3a;  +  1.3a;-l      2  a;  + 1 
a;  —  1         a;4-l         a;  —  2         x-\-2' 

12.    ^^, + ^ +  <^ 


(a  —  6)  (a  —  c)      (6  —  c)  (6  —  a)      (c  —  a)  (c  —  t) 


FRACTIONS 


161 


2     \(m?-^-m 


VK 


m-2\ 
■A-m    J 


^*  \y^    y     JW    y     J 

^^  Va-6     a  +  6     a2  +  62J    g  6^  * 

\a  —  6     a^  —  by  \a  —  b     a/  —  by 


21.    (x^-y^-z^-2yz) 


x-\-y-^z 
x-{-y  —  z 


•    [l-a^l-{-a''^l-a'     1  +  aJ  '  [l -a'^  l-{-a'     l  +  a'J 

g  V     abx^     aca^     b^x     a^x     a 

bd       c^d        d^       cd^      be      d 

cix_b 

c      d 

\  ^        ^^x-\-3yJ     \  ^     x+SyJ 


x-\-Sy, 

25.    /^m-3.y       _4n   \      /m  15nN 

\m-{-nJ\       m-\-nJ      \n  m  J 


26)    1 


2x  +  Bx' 


'  a?  -\-  X 


fSj-Sx\ 
2(a:  +  iy        ^^     2   V(^Tl?/ 

27,    A    J      a;    \/    a;         2x'-\-2ax—a\ 
^    \       a  —  xJ\x-\-a     a^  +  3ax-\-  2  ay 


ACAD.  ALG.  —  11 


162  ACADEMIC  ALGEBRA 

Expand : 

Simplify : 

ii)           f      f .                                  38.    "  +  -\^  +  -^. 
1  +  -  +  -, 


V  J\         X      0^ 


(37. 


g^  4-  3  g  +  2 


m-^  +  yi-^  a2  +  7a  +  12 


+ 


m"  —  n 


a^  +  5  g  H-  4 
3 


(g  4- 1)'  ■  {a  -\-  Vf  r^    m^  +  n^ 


g        ■        1  '      -J 2m7i 

(g  +  1)*      (g  +  1)*  m2  -  mn  +~^ 

1                                    J  1 

^     41.      r, 


1 I 


4 


l-ic  6 


a; 


>l  42.    /^  +  y  I  a^-y\  .  /   a^  +  y  a;-y    Y 

/^N.    ya;^     0?     g     gW^^     a;     ay 

^  aV.,x\ •   . 

^V        gj 
0--^  !  +  _!_ 

r    ,,    a^y    0?"^'    ..  a?.y-i 


44.    -^-^ i-5-5  X  ^ ^-r-  X 


a^y^-xY     i^2__^_i_     xy  +  1 
xy      a^y^ 


SIMPLE    EQUATIONS 


ONE  UNKNO^WN    NUMBER 

186.  An  Equation  has  been  defined,  §  2,  as  an  expression  of 
equality  between  two  numbers  or  quantities. 

187.  An  equation  all  of  whose  known  numbers  are  expressed 
by  figures  is  called  a  Numerical  Equation. 

188.  An  equation  one  or  more  of  whose  known  numbers  is 
expressed  by  letters  is  called  a  Literal  Equation. 

189.  An  equation  that  does  not  involve  an  unknown  number 
in  any  denominator  is  called  an  Integral  Equation. 

2  X 

X  +  6  =  8  and  —  +  5  =  8  are  integral  equations.     The  second  equation 

o 

is  integral ;  for  thougli  it  contains  a  fraction,  the  unknown  number  x  does 
not  appear  in  the  denominator. 

190.  An  equation  that  involves  an  unknown  number  in  any 
denominator  is  called  a  Fractional  Equation. 

8  2  X 

X  +  5  =  -  and  —  7  are  fractional  equations. 

X  x-\ 

191.  An  equation  whose  members  are  identical,  or  such  that 
they  may  be  reduced  to  the  same  form,  is  called  an  Identical 
Equation,  or  an  Identity. 

a-\-h=a-\-h  ]  a^  -  b^  =  {a  +  b)(a  —  b)  are  identical  equations. 

An  equation  whose  members  are  numerical  is  evidently  an 
identical  equation. 

10  =  6  +  4;  8x2  =  6  +  12  —  2  are  identical  equations. 

163 


164  ACADEMIC  ALGEBRA 

A  literal  equation  that  is  true  for  all  values  of  the  letters  in- 
volved is  an  identical  equation,  or  an  identity. 

(x  +  y)2  =  x2  +  2  a;y  +  1/2  jg  ^n  identity,  because  it  is  true  for  all  values  of 
X  and  y. 

192.  An  equation  that  is  true  for  only  certain  values  of  its 
;  letters  is  called  an  Equation  of  Condition. 

I     An  equation  of  condition  is  usually  termed  simply  an  Equation. 
X  +  4  =  10  is  an  equation  of  condition,  because  it  is  true  only  when  the 
value  of  X  is  6.     x^  =  9  is  an  equation  of  condition,  because  it  is  true  only 
when  the  value  of  x  is  +  3  or  —  3. 

^  193.  When  an  equation  is  reduced  to  an  identity  by  the  substi- 
tution of  certain  numbers  for  the  unknown  numbers,  the  equation 
is  said  to  be  satisfied. 

When  X  =  2,  the  equation  3  x  -{-  4  =  10  becomes  6  +  4  =  10,  an  identity  ; 
consequently,  the  equation  is  satisfied. 

Any  number  that  satisfies  an  equation  is  called  a  Root  of  the 
equation. 

2  is  a  root  of  the  equation  3  x  -f  4  =  10. 

Finding  the  roots  of  an  equation  is  called  solving  the  equation. 

194.  An  integral  equation  that  involves  only  the  first  power  of 
one  unknown  number  in  any  term  when  the  similar  terms  have 
been  united  is  called  a  Simple  Equation,  or  an  Equation  of  the  First 
Degree. 

3  X  -f  4  =  10  and  x  +  2y—z  +  8  are  simple  equations. 

195.  Two  equations  that  have  the  same  roots,  each  equation 
having  all  the  roots  of  the  other,  are  called  Equivalent  Equations. 

By  the  axioms  in  §  74,  if  the  members  of  an  equation  are 
equally  increased  or  diminished  or  are  multiplied  or  divided  by 
the  same  or  equal  numbers,  the  resulting  numbers  are  equal  and, 
§  186,  form  an  equation.  But  it  does  not  necessarily  follow  that 
the  equation  so  formed  is  equivalent  to  the  given  equation. 

For  example,  if  both  members  of  the  equation  x  +  2  =  5,  whose  only  root 
is  X  =  3,  are  multiplied  by  x  -  1,  the  resulting  numbers  (x  +  2)(x  -  1)  and 
5(x  —  1)  are  equal  and  form  an  equation  (x  -f-  2)  (x  —  1)  =  5(x  —  1)  ;  but  this 
equation  is  not  equivalent  to  the  given  equation,  since  it  is  satisfied  by  x  =  1 
as  well  as  by  x  =  3. 


SIMPLE  EQUATIONS  165 

196.  Principles.  —  1.  -?/"  the  same  expression  is  added  to  or 
subtracted  from  both  members  of  an  equation,  the  resulting  equation 
is  equivalent  to  the  given  equation. 

2.  If  both  members  of  an  equation  are  multiplied  or  divided  by 
the  same  known  number,  except  zero,  the  resulting  equation  is  equiva- 
lent to  the  given  equation. 

3.  If  both  members  of  an  integral  equation  are  midtiplied  by  the 
same  unknown  integral  expression,  the  resulting  equation  has  all  the 
roots  of  the  given  equation  and  also  the  roots  of  the  equation  formed 
by  placing  the  multiplier  equal  to  zero. 

It  follows  from  Prin.  3  that  if  the  same  unknown  factor  is  removed 
from  both  members  of  an  equation,  the  resulting  equation  has  all  the  roots 
of  the  given  equation  except  those  obtained  from  the  equation  formed  by 
placing  the  factor  removed  equal  to  zero. 

Principle  1  may  be  established  as  follows : 

Let  A  =  B  (1) 

be  any  equation  and  C  any  expression  to  be  added  or  subtracted. 

It  is  to  be  proved  that  A±  C=  B  ±  C  (2) 

is  equivalent  to  (1),  the  given  equation. 

All  the  values  of  the  unknown  number  or  numbers  that  satisfy  (1),  that 
is,  make  A  identical  with  J5,  make  A  +  C  identical  with  B  -\-  C  and  A—  C 
identical  with  B  —  C,  that  is,  satisfy  (2).     Hence,  (2)  has  all  the  roots  of  (1). 

For  the  same  reason     A±Ct  C  =  B±  CT  C  (3) 

has  all  the  roots  of  (2).  But  by  the  Associative  Law  for  addition  (8)  may  be 
written  A  -\-  Q  =  B  +  ^,  or  A  =  B.     Hence,  (1)  has  all  the  roots  of  (2). 

Since  (2)  has  all  the  roots  of  (1)  and  (1)  has  all  the  roots  of  (2),  (2)  is 
equivalent  to  (1),  the  given  equation. 

Principles  2  and  3  may  be  established  as  follows : 

Let  A  =  B  (1) 

and  MA  =  MB.  (2) 

From  (1),  by  Prin.  1,  A-B  =  (i.  (3) 

From  (2),  by  Prin.  1,    M(,A  -B)=0.  (4) 

Since  the  first  member  of  (4)  can  reduce  to  zero  only  when  one  or  both  of 
its  factors  become  0,  (4)  is  satisfied  by  those  values  of  the  unknown  number 
that  make  A  —  B  =  0,  that  is,  by  the  roots  of  (3),  or,  Prin.  1,  of  (1);  and 
also  by  those  values  of  the  unknown  number  that  make  3f  =  0,  that  is,  b3/ 
the  roots  of  M=0,  but  by  no  other  values. 

If  ilf  is  any  known  number,  not  zero,  M  cannot  be  placed  equal  to  zero 
and  then  (4),  or  (2),  is  equivalent  to  (3),  or  to  (1). 


166  ACADEMIC  AL.GEBRA 

If  Mia  an  unknown  expression,  (4),  or  (2),  has  the  roots  of  il^=  0  in  addi- 
tion to  the  roots  of  ^1  —  ^  =  0,  or  of  (1). 

197.  By  §  196,  Prin.  1  and  2,  every  simple  equation  involving 
one  unknown  number  may  be  reduced  to  an  equivalent  equation 
having  the  form  x  =  a^  a  being  a  fixed  known  number.     Hence, 

Every  simple  equation  involving  one  unknown  number  has  one 
root  and  only  one  ;  also,  by  §  196,  Prin.  3, 

Tlie  equation  (x  —  a)(x  —  b)  (x  —  c)  "•  (x  ~  r)  =  0  is  equivalent 
to  the  simple  equations  x—a  =  0,  x  ~b  =  0,  x  —  g=  0,  ••-  x  — 7'  =  0, 
and  has  as  many  roots  as  factors  of  the  first  degree  iyivolving  x. 

CLEARING   EQUATIONS    OP   FRACTIONS 

198.  1.    If  one  third  of  a  number  is  10,  what  is  the  number  ? 

2.  li  \x  =  l,  what  is  the  value  of  x?  If  ia;  =  5,  what  is  the 
value  of  07  ?     If  i  a;  =  6,  what  is  the  value  of  a;  ? 

3.  If  ^x^Q>,  what  is  the  value  of  2a;?  If  ^x  =  10,  what  is 
the  value  of  5  a;  ? 

X 

4.  If  -  =  5,  what  is  the  form  of  the  resulting  equation  when 

o 

both  members  are  multiplied  by  3  ?  by  6  ?  by  9  ?  by  any  multiple 
of  the  denominator  ? 

199.  The  process  of  changing  an  equation  containing  fractions 
to  an  equation  without  fractions  is  called  Clearing  the  Equation  of 
Fractions. 

200.  Principles.  —  1.  An  equation  may  be  cleared  of  fractions 
by  multiplying  both  members  by  some  common  multiple  of  the  denomi- 
nators of  the  fractions.     (Ax.  4.) 

2.  If  both  members  of  a  fractional  equation  are  multiplied  by  the 
expression  of  lowest  degree  required  to  clear  the  equation  offractioris, 
the  resulting  equation  is  equivalent  to  the  given  equation. 

Principle  2  may  be  established  as  follows : 

By  §  196,  Prin.  1,  all  the  terms  of  the  second  member  may  be  transposed 
to  the  first  member.  Hence,  uniting  the  terms  of  the  first  member  into  one 
and  reducing  this  to  its  lowest  terms,  any  fractional  equation  may  be  reduced 
to  an  equivalent  equation  of  the  form 

^  =  0,  (1) 


SIMPLE   EQUATIONS  167 

in  which  A  is  prime  to  jB,  and  B  is  the  expression  of  lowest  degree  required 
to  clear  the  equation  of  fractions. 

Since  A  and  B  have  no  common  factors,  A  and  B  cannot  reduce  to  zero 
at  the  same  time  for  any  value  of  the  unknown  number.  Hence,  eq.  (1) 
is  satisfied  by  every  value  of  the  unknown  number  that  makes  ^  ==  0,  and  by 
no  other  values  ;  that  is,  the  equation  ^  =  0,  obtained  by  multiplying  both 
members  of  the  given  equation  by  the  expression  of  lowest  degree  required 
to  clear  it  of  fractions,  is  equivalent  to  the  given  equation. 

.  Examples 

1.  Given  — — —  =6 ,  to  find  the  value  of  x. 

4  3' 

PROCESS 

a:  —  4      n      X 

Cleajing  of  fractions,  3a;  —  12  =  72  —  4a; 

.-.  a;  =12 

Explanation.  —  Since  the  first  fraction  will  become  an  integer  if  the 
members  of  the  equation  are  multiplied  by  4  or  any  multiple  of  4,  and  since 
the  second  fraction  will  become  an  integer  if  the  members  of  the  equation  are 
multiplied  by  3  or  any  multiple  of  3,  the  equation  may  be  cleared  of  fractions 
in  a  single  operation  by  multiplying  its  members  by  some  common  multiple 
of  4  and  3  (§  196,  Prin.  2,  §  200,  Prin.  1). 

Since  the  terms  derived  from  the  numerators  will  be  least  or  lowest  when 
the  multiplier  is  the  least  or  lowest  common  multiple  of  the  denominators, 
the  members  of  the  equation  should  be  multiplied  by  the  least  common  mul- 
tiple of  4  and  3,  which  is  12. 

The  resulting  equation  is  3  x  —  12  =  72  —  4  a; ;   .-.  a;  =  12. 

2.  Find  the  value  of  x  in  ^— ^^^^ ^^^—  = • 

2  3         3         4 

e  '  x-lx-22a;-3 

Solution. —  = 

2  3  3         4 

Clearing  of  fractions,  6(x  -  1)  -  4(x  -  2)  =  8  -  3(a;  -  3). 

6x-G-4x  +  8  =  8-3x  +  9. 

.-.  X  =  3. 


168      -  ACADEMIC  ALGEBRA 

Rule.  —  Multiply  both  members  of  the  equation  by  the  least  or 
lowest  common  multiple  of  the  denominators. 

1.  The  multiplier  of  lowest  degree  required  to  clear  an  equation  of  frac- 
tions so  that  the  resulting  equation  is  equivalent  to  the  given  equation  is 
usually  the  L.  C.  D.  But  if  fractions  having  like  denominators  have  not 
been  united  and  every  fraction  reduced  to  its  lowest  terms,  the  multiplier 
required  may  be  of  lower  degree  than  the  L.  C,  D. 


Thus,  given 

2a;2  _3a:  +  2      bx  +  % 
x-2       x-2            3 

Uniting  terms,   ^^'^-^^- 
X  —  2 

-2,or2x  +  l=^^  +  ^. 
3 

Multiplying  by  3, 

6x  +  3  =  5x  +  8. 

.-.  x  =  b. 

Had  the  given  equation  been  cleared  of  fractions  by  multiplying  by 
3(x  —  2)  instead  of  by  3,  the  resulting  equation  simplified,  which  is 
x'"^  —  7  X  +  10  =  0,  or  {x  —  5)  (x  —  2)  =  0,  would  have  been  satisfied  both  by 
X  =  5  and  by  x  =  2.  The  given  equation  is  satisfied  by  x  =  5  but  not  by 
X  =  2.  Hence,  the  latter  value  is  not  a  root  of  the  given  equation,  but  has 
been  introduced  by  using  (x  —  2)  times  the  necessary  multiplier,  3.  Roots 
so  introduced  may  be  discovered  by  verification  and  rejected. 

2.  If  a  fraction  has  the  minus  sign  before  it,  the  signs  of  all  the  terms 
of  the  numerator  must  be  changed  when  the  denominator  is  removed. 

Find  the  value  of  x,  and  verify  the  result : 

3.  2a;-f-  =  55. 

3       3 

4.  ^  +  10  =  13. 
4 


5.   •^-f-2x  =  26. 
6 


6.    3a;-^  =  14. 


^^        X    .    X        X    .    rj  X 

'    2'^3~'4"^10 


^12. 


£(y_x      i)  X      'zx 
18'       9        3 


7. 

XX     10 
2     6      3' 

8. 

rri         SX        X 

'^      14  "7* 

9. 

1^-1=''' 

10. 

2x     5_x 
3       6     4* 

nx 

12' 

=  7. 

5a; 

o 

1  q      ?_^  _^"'_^i'^^_  ^ 


SIMPLE   EQUATIONS  169 

14  ^^   7.r5a;   a;_4 

3  8   18   24~9* 

15  3a;   7a;   a;.9a;_l 

4  16   ^   16~8* 

-_  15  a;  ,  5  a;   11a;  ,  19  a;   « 
7    6    3    14 

17  2  a;   5a;   4a;  a;_  1 

'  15   25   9   6~9" 

18  ^_I-?  =  1L^_^4-? 

*  4   12  "  36    9   2* 

19  ^J_JL  =  ?^_|_!^4.ij^ 
'  4  "^20   10  "^5"^  15* 

20.  I:^-2a;  +  i^i:I  =  l. 


--  a;  — 1  ,  a;  — 2  ,  a;  —  3   5a;  — 1 

^'-  ^T~+^~  +  ^r="~6— 

22.  -^ _  +  ^_16. 

7a;  +  2   12-a;  a;  +  2_^ 

23.  -^^ _  +  ___6. 

24  «^-3,a.'  +  5   a;4-2^^ 
'   7     3     6 

25.  >2iL-j_7.^-13^3_^+3, 

4  6         2 

26.  ^^-'"^   3a;-2^^   a;  +  2^ 

5  7         6 

l_2a;   7-2a;  ll-2a;_   7 
3       4   "^   6      12^ 

28  ^  +  4   2-2a;_a;  +  l   oi 


170  ACADEMIC   ALGEBRA 

14      "^6a.'  +  2  50  14° 

Suggestion.  — The  equation  may  be  written 

§104,3,  '  9^  +  A  +  8^^  =  '36£^15      41, 

14       14      6  a:  4-  2       66       56      56 

Simplify  as  much  as  possible  before  clearing  of  fractions. 

30  3  a; -2      3  a;  -  21  ^  6  a;  -  22 

'    2x-b  5  10      * 

31  4  g;  +  3  ^  8  g;  +  10      7  a;  -  29 

9       ~       18  5a; -12* 

6a;  +  l       2a;-4       2a;-l 


32. 


33. 


34. 


35. 


15  7  a;  - 13           5 

10a;H-17      5.T-2^12a;  -1 

18  9       ~lla;-8' 

6x  +  3  3a;-l  _2a;-9 

15  5a;-25~       5 

2a;  +  l  8       ^2a;-l 

2a;_l  4a;^-l      2a;H-l' 


36.    Solve  the  equation   ^^:il  +  ^^  =  ^^  +  ^^. 
^  a;-2a;-7      a;-6a;-3 

SoLUTiov.  —  It  will  be  observed  that  if  the  fractions  in  each  member  were 
connected  by  the  sign  — ,  and  the  members  were  simplified,  the  numerators 
of  the  resulting  fractions  would  be  simple.  The  fractions  can  be  made  to 
meet  this  condition  by  transposing  one  fraction  in  each  member. 

Consequently,  it  is  sometimes  expedient  to  defer  clearing  of  fractions. 

Transposing,  ^~  ^  -  ^^^  _  ^-^  _  ^  ~-  ^\ 

x-2     a;  —  3     x-(>     x-1 

Uniting  terms,  ~  ~ 


x2_5a;  +  6     ic-^-13x  +  42 

Since  the  fractions  are  equal  and  their  numerators  are  equal,  their  denomi- 
nators must  be  equal. 

a:'^  -  5  X  +  6  =  cc2  -  13  x  +  42. 


37. 
38. 
39. 
40. 
41. 

42. 
43. 
44. 
45. 

46. 

47. 
48. 

49. 
50. 

51. 


SIMPLE  EQUATIONS 

X  —  1        X  —  1  _X  —  ^        X  —  Z 

X  —2      X  —  8      X  —  6      X  —  4: 

X—  3      X  —  7  _x  —  6     X  —  4: 
X  —  4:      X  —  S      X  —  7     X  —  5 

x-{-2  _  x  +  S  __  a;  +  5  _  x-\-6 
x-{-l      x-{-2~  x  +  4:      x-fS' 

2      re  +  o 


171 


X  -^1      X  -^  6  _x 


x  +  2      x  +  7      x-^3     x-\-6 

x  —  5      a;  —  10_a;  —  4      x  —  d 
ic-fo      x-\-10~  x-\-4:      a- +  9* 


<  X 
3 


2     5x- 


2      4 


-2 


4 
x-1 


3 


-3i. 


x—2      x—3     X 


x  —  1       x-\-l 
ar3  4-  2      ar^  -  2 


8 


ar'-l 
10 


x  -\-l       X- 

2^  +  4 
3  ^         7. 


1      ar-1 


-3  =  0. 


le-) 


16     --4-6 

5.^ 41. 


24  60     ~  5 

^•(2-a.)-f(3-2a.)=^±12. 


K'-9 


3a;fl 


+ 


'4x 


--ie-^0' 


(2  a;  4- 1)^      (4  a;  -  1)^  ^  15      3(4  .t  -f  1) 
5  20  8  40       ' 


1T2  ACADEMIC  ALGEBRA 


52.    l.^ii-|-2  =  ^+^-^ 


2a;-l  2ic  +  l      1-ar^ 

17+?      1+1?      2-1-1      1^  +  ? 

54.    i(a^  -  4)      4  .T  -  16  _  3       5 


2  6  5  I 


10 


,           1          '+x  1 

55.    iH ^  = 7^-  56. 


1  +1       1+-  *  -—1         ^"^^ 

X  X  a?     3 


3      3a;-l 


1,    Solve  the  equation 


Literal  Equations 
X  —  b'      X  —  a' 


a  b 

Solution 
X  —  b^     X  —  a 


a  b 

Clearing  of  fractions,  bx  —  b^  =  ax  —  a*. 

Transposing,  etc.,  ax  —  bx  =  a^  —  b^. 

(a  —  h)x  =  a^  —  b^. 
Dividing  by  (a- 6),  x  =  a- +  ab -\- b^. 

Solve  the  following  equations : 

2     ^ ~  ^  I  ^^ _  1 .  f,     X  —  2ab     1  _x  —  3c 

nx        ex     c  '         ex  X         abx 

^^    x-a  ^_2x^^  ^  6b 
b  a  a 

8     «'  I    b'  ^a-\-b      S(a  +  b) 
bx     ax        ab  x 

^     a2  +  &2  _  a-b  ^  b^ 
2bx    '     2bx'      X 


3. 

1- 

ab_l_ 
X      ab 

49 
abx 

4. 

a^ 

aW 

2a' _^      2b\ 
y^x             a'x 

5. 

X 

b 

x  +  2b  _ 
a 

-I-'- 

SIMPLE  EQUATIONS  173 

in     r,   \  ^(^ -{■  a)  _  2  ah  x  —  2a  .  x _a-  -\-W 

X  —  a       X—  a  a  h         ah  ^ 

12.  ^x^- 18^1  - 1^  =  a{x  -  a). 

13.  6(2ic  -  9c  -  14 /^)=c(c- it'). 

14.  a{x  —  a  —  2h)-\-h{x  —  h)-\-c{x^c>)=  0. 

15.  (a  —  x){x  —  6)  +  (a  +  a;)(a;  —  6)  =  (ct  —  6)-. 
a  —  h  -\-  c      h  —  a  -\-  c 


16 


ic  +  a  a;  —  a 


17. H — L_  =  o. 

a(6  —  x)      h{c  —  x)      a(c  —  x) 

37—  1       a  —  \  7?  —  c? 


a-\      x-\      (a-  \){x  -  1) 

a  ■\-  X        2x        Qi?{x  —  a)  _  1 
a         a  -\-  X      a{a^  —  x^)     3 

20.    ^-±^  +  ^±^+^±^  =  "  +  ^  +  ^4-1. 
h  a  c       ■  h      c     a 

^2      a^—  ax—  bx  -^ab  _x^  —  2hx  -^2h^ 


22. 


x  —  a  X  —  b  X  —  c 

1  2  mn  m  x  —  n       . 


m  -{-  n      (m  +  ny      (m  +  itf      (m  -|-  n)^ 


23.    ^^ 4- "^ =a^-^h''  +  c'  +  2ab. 

a-\-  b  +  c     a  -\-h  —  c 


V^XrV 


x-{-2h,x  +  ?>h       x  +  b    ,  x  +  2h 


:2^^^^^^^-^^^r^    2a;  +  3a^3a;  +  7a^      2a       ^  ^ 
x  -\-  a  x  -\-2  a       a;H-4a 

X  -{-1  a       X  —  a  _x  -\-  1  a       x  —  a 


X  -\-  Q  a     X  —  ^  a       X  -\-  a       x  -\-2  a. 
27.    (a  —  h)(x  —  c)  —  (b  —  c)(x  —  a)  =  (c  —  a)(x  —  b). 


174  ACADEMIC  ALGEBRA 


Problems 


Directions  for  Solving.  —  Represent  one  of  the  unknown 
numbers  by  x,  and  from  the  conditions  of  the  problem  find  an 
expression  for  each  of  the  other  unknown  numbers. 

Find  from  the  problem  two  expressions  that  are  equal,  and  write 
them  as  an  equation. 

Solve  the  equation. 

1 .  A  man  bought  a  farm,  a  house,  and  a  barn  for  $  12,600. 
If  the  house  cost  twice  as  much  as  the  barn,  and  the  farm  twice 
as  much  as  the  house  and  barn  together,  how  much  did  each 
cost? 

Suggestion. — There  are  three  unknown  quantities  —  the  cost  of  the 
house,  the  cost  of  the  barn,  and  the  cost  of  the  farm.  It  is  evident  that,  if 
the  cost  of  the  barn  were  known,  the  cost  of  the  house  could  be  found  from 
it,  and  from  both  the  cost  of  the  farm. 

Accordingly,  represent  the  cost  of  the  barn  as  x  dollars,  and  express  the 
cost  of  the  house  and  the  cost  of  the  farm  in  terms  of  x.  Discover  two 
expressions  for  the  total  cost,  and  equate  them. 

2.  If  A  is  twice  as  old  as  B,  and  B  twice  as  old  as  C,  how 
old  is  each, if  the  sum  of  their  ages  is  140  years  ? 

3.  Mr.  Henry  bought  three  building  lots  for  ^36,000.  If  the 
third  cost  twice  as  much  as  the  second,  and  the  second  3  times 
as  much  as  the  first,  what  was  the  cost  of  each  ? 

4.  A  man  left  ^  63,000  to  his  wife,  two  sons,  and  a  daughter. 
If  each  son  received  twice  as  much  as  the  daughter,  and  half  as 
much  as  the  wife,  what  was  the  share  of  each  ? 

5.  A  man  left  \  of  his  property  to  his  wife,  \  to  his  son,  4  to 
his  daughter,  and  the  rest,  which  was  ^2000,  to  a  hospital.  What 
was  the  value  of  his  property  ? 

6.  A  owed  B,  C,  and  D  ^  27,000  in  all.  If  he  owed  B  4  times 
as  much  as  C,  and  D  |  as  much  as  B,  what  sum  did  he  owe 
each  ? 

7.  A  person  spends  \  of  his  annual  income  for  board,  \  for 
clothes,  and  $260  for  other  expenses.  If  he  saves  \  of  his 
income,  what  is  his  income? 


SIMPLE   EQUATIONS  175 

^8.    A  table  and  a  chair  cost  f  11.     The  table  and  a  picture 

cost  $  14.     If  the  chair  and  the  picture  together  cost  3  times  as 
much  as  the  table,  what  was  the  cost  of  each  article  ? 

9.  A  and  B  together  had  $  50,  and  A  and  C  had  $  60.  After 
each  had  spent  ^5,  A  had  \  as  much  as  B  and  C  together. 
How  much  had  each  at  first? 

10.  A  gave  his  age  as  follows :  "|  of  my  age  less  i  of  what  it 
will  be  a  year  hence  is  equal  to  \  of  my  age  5  years  ago."  What 
was  his  age  ? 

11.  James  is  5  years  older  than  his  sister,  and  5  years  hence 
he  will  be  3^  times  as  old  as  his  sister  was  5  years  ago.  What  is 
the  age  of  each  ? 

12.  A's  daily  wages  are  |  of  B's,  and  C's  are  |  of  A's.  If  A 
and  C  together  earn  25  cents  more  a  day  than  B,  what  are  the 
daily  wages  of  each  ? 

13.  A  man  paid  a  debt  of  $8.00  with  an  equal  number  of 
5,  10,  and  25-cent  pieces.     How  many  of  each  were  there  ? 

14.  A  man  bought  equal  quantities  of  white  and  brown  sugar, 
paying  6^  cents  a  pound  for  the  former  and  5  cents  a  pound  for 
the  latter.  How  many  pounds  of  each  did  he  buy,  if  the  whole 
quantity  cost  him  $  1.80  ? 

15.  A  field  is  twice  as  long  as  it  is  wide.  By  increasing  its 
length  20  rods  and  its  width  30  rods,  the  area  will  be  increased 
2200  square  rods.     What  are  its  dimensions  ? 

16.  Three  fifths  of  a  certain  number  exceeds  \  of  it  by  7. 
What  is  the  number? 

17.  One  third  of  a  number  added  to  3  times  the  number  is 
equal  to  50.     What  is  the  number  ? 

18.  After  spending  f  of  my  income  and  $300  more,  I  had 
\  of  it  left.     What  was  my  income  ? 

19.  The  sum  of  J  of  a  number,  \  of  it,  and  J  of  it  is  4  more 
than  f  of  the  number.     What  is  the  number  ? 

20.  The  sum  of  a  number,  its  half,  its  third,  and  its  fourth, 
and  16  is  66.     What  is  the  number  ? 


176  ACADEMIC  ALGEBRA 

21.  A  man  deposited  J  of  his  month's  wages  in  a  bank,  and 
paid  out  i  of  the  remainder  for  groceries  and  $  9  for  dry  goods. 
If  he  had  $  3  left,  how  much  money  had  he  at  first  ? 

22.  Divide  500  into  two  parts,  such  that  the  greater  decreased 
by  \  of  the  smaller  is  5  times  as  much  as  the  smaller  decreased 
by  ^  of  the  larger. 

23.  Divide  61  into  two  parts,  such  that,  if  the  greater  is 
divided  by  the  less,  the  quotient  will  be  3  and  the  remainder  1. 

24.  Divide  111  into  two  parts,  such  that  the  first  diminished 
by  3  is  equal  to  the  second  divided  by  3. 

,^-^5.   Divide  40  into  two  parts,  such  that  the  first  divided  by  3 
is  equal  to  the  second  divided  by  5. 

26.  Divide  40  into  two  parts,  such  that  the  first  is  10  greater 
than  twice  the  second. 

27.  Divide  44  into  two  parts,  such  that,  if  the  greater  is  divided 
by  5  and  the  less  by  7,  the  difference  of  the  quotients  will  be  4. 

28.  Divide  54  into  two  parts,  such  that  ^  of  the  first  part  is 
equal  to  I  of  the  second. 

Solution 
Let  X  =  \  oi  the  first  part  or  ^  of  the  second  part. 

Then,  4  a;  =  the  first  part, 

and  6x  =  the  second  part ; 

4x  +  5a;  =  54; 
whence,  a;  =  6, 

4  ic  =  24,  the  first  part, 
and  5  a;  =  30,  the  second  part. 

29.  Divide  40  into  three  parts,  such  that  ^  of  the  first,  ^  of  the 
second,  and  I  of  the  third  are  equal. 

30.  Find  three  parts  of  60,  such  that  the  first  divided  by  5. 
the  second  multiplied  by  2,  and  the  third  increased  by  5  are 
equal. 

31.  Divide  72  into  four  parts,  such  that  the  first  divided  by  2, 
the  second  diminished  by  2,  the  third  multiplied  by  2,  and  the 
fourth  increased  by  2  are  equal. 


SIMPLE  EQUATIONS  177 

32.  A  can  do  a  piece  of  work  in  8  days.  If  B  can  do  it  in  10 
days,  in  how  many  days  can  both  working  together  do  it  ? 

Solution 
Let  X  =  the  required  number  of  days. 

Then,  -  =  the  part  of  the  work  both  can  do  in  1  day, 

^  =  the  part  of  the  work  A  can  do  in  1  day, 
^  =  the  part  of  the  work  B  can  do  in  1  day ; 

a;     8      10     40* 
Solving,         x  =  ^,  or  4|,  the  required  number  of  days. 

33.  A  can  do  a  piece  of  work  in  10  days,  and  B  can  do  it 
in  15  days.     How  long  will  it  take  both  to  do  it? 

34.  Three  pipes  empty  into  a  cistern.  One  can  fill  the  cistern 
in  5  hours,  another  in  6  hours,  and  the  third  in  10  hours.  How 
long  will  it  take  the  three  pipes  together  to  fill  it  ? 

^%h.  A  can  do  a  piece  of  work  in  10  days,  B  can  do  it  in  12 
days,  and  C  can  do  it  in  8  days.  In  how  many  days  can  all 
together  do  it? 

36.  A  can  pave  a  walk  in  6  days,  and  B  can  do  it  in  8  days. 
How  long  will  it  take  A  to  finish  the  job  after  both  have  worked 
3  days  ? 

37.  A  can  build  a  wall  in  15  days,  but  with  the  aid  of  B  and 
C,  the  wall  can  be  built  in  6  days.  If  B  does  as  much  work 
in  1  day  as  C  does  in  2  days,  in  how  many  days  can  B  and  C 
separately  build  the  wall? 

V  38.  A  and  B  can  dig  a  ditch  in  10  days,  B  and  C  can  dig  it  in 
6  days,  and  A  and  C  in  1\  days.  In  what  time  can  each  man  do 
the  work  ? 

Suggestion.  —  Since  A  and  B  can  dig  ^^  of  the  ditch  in  1  day,  B  and  C 
\  of  it  in  1  day,  and  A  and  C  ^  of  it  in  1  day,  ^  +  i  +  i^^  is  twice  the  part 
they  can  all  dig  in  1  day. 

39.  A  and  B  can  load  a  car  in  2^  hours,  B  and  C  in  3f  hours, 
and  A  and  C  in  3jij  hours.  How  long  will  it  take  each  alone  to 
load  it  ? 

ACAD.   ALG.  12 


178  ACADEMIC   ALGEBRA 

40.  A  boy  bought  some  oranges  at  the  rate  of  30  cents  a  dozen. 
He  sold  J  of  them  for  4  cents  each,  and  the  rest  for  3  cents  each. 
If  he  gained  90  cents,  how  many  oranges  did  he  buy  ? 

41.  Find  a  fraction  whose  value  is  ^  and  whose  denominator 
is  15  greater  than  its  numerator. 

42.  Find  a  fraction  whose  value  is  f  and  whose  numerator  is 
3  greater  than  half  of  its  denominator. 

^""  43.  The  numerator  of  a  certain  fraction  is  8  less  than  the  de- 
nominator; and  if  each  term  of  the  fraction  is  decreased  by  5, 
the  value  of  the  fraction  becomes  |.     What  is  the  fraction  ? 

44.  The  units'  digit  of  a  number  expressed  by  two  digits  ex- 
ceeds the  tens'  digit  by  5.  If  the  number  increased  by  63  is 
divided  by  the  sum  of  its  digits,  the  quotient  is  10.     What  is  the 

number  ? 

Solution 

Let  X  =  the  digit  in  tens'  place. 

Then,  x+  6  =  the  digit  in  units'  place, 

and  10  aj  -f  (aJ  +  5),  or  11  oj  -f  5  =  the  number  ; 

11  a;  +  5  4-  63  ^  ^^ . 
2x  +  5 
whence,  x  =  2, 

and  a;  +  5  =  7. 

Therefore,  the  number  is  27. 

45.  The  tens'  digit  of  a  number  expressed  by  two  digits  is 
3  times  the  units'  digit.  If  the  number  diminished  by  33  is 
divided  by  the  difference  of  the  digits,  the  quotient  is  10.  What 
is  the  number  ? 

"^46.  The  tens'  digit  of  a  number  expressed  by  two  digits  is  ^ 
of  the  units'  digit.  If  the  number  increased  by  27  is  divided  by 
the  sum  of  its  digits,  the  quotient  is  6^.     What  is  the  number  ? 

47.  In  a  purse  containing  |>1.45  there  are  ^  as  many  quarters 
as  5-cent  pieces  and  -|  as  many  dimes  as  5-cent  pieces.  How 
many  pieces  are  there  of  each  kind  ? 

^48.  A  woman  spent  $10  more  than  f  of  her  money;  then 
$.10  more  than  |  of  the  remainder.  If  she  had  $2  left,  liow 
much  money  had  she  at  first  ? 


SIMPLE  EQUATIONS  179 

49.  A  man  spent  $1  less  than  |  of  his  money  and  had  left 
^  1  less  than  J  of  it.     How  much  money  had  he  at  first  ? 

50.  A  girl  found  that  she  could  buy  12  apples  with  her  money 
and  have '5  cents  left,  or  10  oranges  and  have  6  cents  left,  or  6 
apples  and  6  oranges  and  have  2  cents  left.  How  much  money 
had  she  ? 

51.  A  boy  spent  \  of  his  money  and  \  a  cent  more,  then  \  of 
the  remainder  and  \  a  cent  more,  then  i  of  what  he  had  left  and 
^  a  cent  more,  when  he  found  that  he  had  2  cents  remaining. 
How  much  had  he  at  first  ? 

52.  Five  boys  bought  a  boat,  agreeing  to  share  the  expense 
equally.  But  one  of  them  having  left  $1  of  his  share  unpaid, 
each  of  the  others  had  to  pay  -^^  more  than  one  fifth  of  the  ex- 
pense.    What  was  the  cost  of  the  boat  ? 

53.  A  sum  of  money  was  divided  among  A,  B,  C,  and  D  so 
that  A  received  \  as  much  as  all  the  others,  B  received  \  as  much 
as  all  the  others,  C  received  \  as  much  as  all  the  others,  and  D 
received  $  2800  less  than  A.     What  sura  did  each  receive  ? 

54.  In  an  alloy  of  90  ounces  of  silver  and  copper  there  are  6 
ounces  of  silver.  How  much  copper  must  be  added  that  10 
ounces  of  the  new  alloy  may  contain  |  of  an  ounce  of  silver  ? 

55.  If  80  pounds  of  sea  water  contain  4  pounds  of  salt,  how 
much  fresh  water  must  be  added  that  45  pounds  of  the  new  solu- 
tion may  contain  If  pounds  of  salt? 

56.  An  officer,  attempting  to  arrange  his  men  in  a  solid  square, 
found  that  with  a  certain  number  of  men  on  a  side  he  had  34  men 
over,  but  with  1  man  more  on  a  side  he  needed  35  men  to  com- 
plete the  square.     How  many  men  had  he  ? 

Suggestion.  —  With  x  men  on  a  side,  tlie  square  contained  o^^  men  ;  with 
X  +  1  men  on  a  side,  there  were  places  for  (ar  +  1  )2  men.  Since  the  number 
of  men  was  the  same  under  both  arrangements,  x^  -f  34  =  (x  +  1)"^  —  35. 

57.  A  regiment  drawn  up  in  the  form  of  a  solid  square  lost  60 
men  in  battle.  Afterward,  when  the  men  were  arranged  in  a 
solid  square  with  1  man  less  on  a  side,  it  was  found  that  there 
was  1  man  over.  How  many  men  were  there  in  the  regiment  at 
first? 


180  ACADEMIC  ALGEBRA 

58.  A  regiment  drawn  up  in  the  form  of  a  solid  square  was 
reenforced  by  240  men.  When  the  regiment  was  formed  again  in 
a  solid  square,  there  were  4  more  men  on  a  side.  How  many 
men  were  there  in  the  regiment  at  first  ? 

59.  A  man  was  hired  for  40  days  under  the  following  condi- 
tions :  for  every  day  he  worked  he  was  to  receive  $  3  besides  his 
-board,  while  for  every  day  he  was  idle  he  was  to  receive  nothing, 
but  was  to  be  charged  $  1.20  for  his  board.  If  at  the  end  of  the 
period  he  received  $  57,  how  many  days  did  he  work  ? 

60.  A  man  invested  $800,  a  part  at  6%  and  the  rest  at  5%. 
If  the  total  annual  interest  was  f  45,  how  much  did  he  invest  at 
each  rate  ? 

Suggestion.  —  Let  x  =  the  number  of  dollars  invested  at  6  %. 
Then,  800  —  x  =  the  number  of  dollars  invested  at  5  % ; 

61.  A  man  has  |  of  his  property  invested  at  4%,  J  at  3%,  and 
the  remainder  at  2%.  How  much  property  has  he,  if  his  annual 
income  is  $  860  ? 

62.  A  man  put  out  $4330  in  two  investments.  On  one  of 
them  he  gained  12%,  and  on  the  other  he  lost  5%.  If  his  net 
gain  was  $  251,  what  was  the  amount  of  each  investment  ? 

63.  There  were  distributed  among  20  men  and  25  women  $  160 
in  such  a  way  that  the  sum  of  what  a  man  and  a  woman  received 
was  $  7.  How  much  did  the  men  receive,  and  how  much  did  the 
women  receive  ? 

64.  At  what  time  between  5  and  6  o'clock  will  the  hands  of  a 
clock  be  together  ? 

Solution 
Let         X  =  the  number  of  minute  spaces  that  the  minute  hand  travels 

after  5  o'clock  before  they  come  together. 
Then,    —  =  the  number  of  minute  spaces  that  the  hour  hand  travels  in 

the  same  time. 
Since  they  are  25  minute  spaces  apart  at  5  o'clock, 

X-  —  =2G: 
12 

.'.  X  =  27^j  the  number  of  minutes  after  5  o'clock. 

65.  At  what  time  between  1  and  2  o'clock  will  the  hands  of  a 
clock  be  together  ? 


SIMPLE  EQUATIONS  181 

66.  At  what  time  between  6  and  7  o'clock  will  the  hands  of  a 
clock  be  together  ? 

67.  At  what  time  between  10  and  11  o'clock  will  the  hands  of 
a  clock  point  in  opposite  directions  ? 

68.  At  what  times  between  4  and  5  o'clock  will  the  hands  of  a 
clock  be  15  minute  spaces  apart  ? 

69.  When  after  9  o'clock  and  before  10  o'clock  will  the  hands 
of  a  clock  be  at  right  angles  to  each  other  ? 

70.  A  man  rows  downstream  at  the  rate  of  6  miles  an  hour 
and  returns  at  the  rate  of  3  miles  an  hour.  How  far  downstream 
can  he  go  and  return  within  9  hours  ? 

71.  At  the  rate  of  3  miles  an  hour  uphill  and  4  miles  an  hour 
downhill  a  woman  can  walk  60  miles  in  17  hours.  How  much 
of  the  distance  is  uphill,  and  how  much  is  downhill  ? 

72.  A  hare  pursued  by  a  hound  takes  4  leaps  while  the  hound 
takes  3 ;  but  2  of  the  hound's  leaps  are  equal  to  3  of  the  hare's. 
If  the  hare  has  a  start  equal  to  60  of  her  own  leaps,  how  many 
leaps  must  the  hound  take  to  catch  the  hare  ? 

Solution 
Let  Sx  =  the  number  of  leaps  taken  by  the  honnd. 

Then,  4  a;  =  the  number  of  leaps  taken  by  the  hare. 

Suppose  a  =  the  number  of  feet  in  one  leap  of  the  hare. 

Then,  —  =  the  number  of  feet  in  one  leap  of  the  hound, 

2 

—  X  3  a;  =  — ^,  the  number  of  feet  the  hound  runs, 
2  2 

and  a  X  4  a;  =  4  ax,  the  number  of  feet  the  hare  runs. 

Since  the  hare  has  a  start  equal  to  60  times  a  feet,  or  60  a  feet,  the  hare 
runs  60  a  feet  less  than  the  hound. 


and 


Therefore, 

4ax  =  ^^-G0a. 

Dividing  by  a. 

4x  =  ^^-60. 

2 

Therefore, 

X  =  120, 

d 

3  X  =  360,  the  number  of  leaps  taken  by  the  hound. 

182  ACADEMIC  ALGEBRA 

73.  A  fox  is  70  leaps  ahead  of  a  hound  and  takes  5  leaps  while 
the  hound  takes  3 ;  but  3  of  the  hound's  leaps  equal  7  of  the 
fox's.     How  many  leaps  must  the  hound  take  to  catch  the  fox  ? 

74.  A  rabbit  makes  5  leaps  while  a  dog  makes  4 ;  but  3  of  the 
dog's  leaps  are  equal  to  4  of  the  rabbit's.  If  the  rabbit  has  a 
start  of  20  leaps,  how  many  leaps  will  each  take  before  the  rabbit 
is  caught? 

75.  A  hound  is  39  of  his  leaps  behind  a  rabbit  that  takes  7 
leaps  while  the  hound  takes  8.  If  6  leaps  of  the  rabbit  are  equal 
to  5  leaps  of  the  hound,  how  many  leaps  must  the  hound  tak6  to 
catch  the  rabbit  ? 

76.  A  wheelman  and  a  pedestrian  leave  the  same  place  at  the 
same  time  to  go  to  a  point  54  miles  distant,  the  former  traveling 
3  times  as  fast  as  the  latter.  The  wheelman  makes  the  trip  and 
returning  meets  the  pedestrian  in  6f  hours  from  the  time  they 
started.     What  is  the  rate  of  each  ? 

77.  If  1  pound  of  lead  loses  -^\  of  a  pound,  and  1  pound  of 
iron  loses  ^-^  of  a  pound  when  weighed  in  water,  how  many  pounds 
of  lead  and  of  iron  are  there  in  a  mass  of  lead  and  iron  that 
weighs  159  pounds  in  air  and  143  pounds  in  water  ? 

78.  If  97  ounces  of  gold  weighs  92  ounces  when  it  is  weighed 
in  water,  and  21  ounces  of  silver  weighs  19  ounces  when  it  is 
weighed  in  water,  how  many  ounces  of  gold  and  of  silver  are 
there  in  a  mass  of  gold  and  silver  that  weighs  320  ounces  in  air 
and  298  ounces  in  water  ? 

79.  A  merchant  increases  his  capital  annually  by  \  of  it,  and 
at  the  end  of  each  year  takes  gut  $  800  for  expenses.  At  the  end 
of  three  years,  after  taking  out  his  expenses,  he  finds  that  his 
capital  is  $  6325.     What  was  his  original  capital  ? 

80.  A  merchant  added  annually  to  his  capital  ^  of  it,  and  at 
the  end  of  each  year  took  out  $  1000  for  expenses.  If  at  the  end 
of  the  third  year,  after  taking  out  the  last  ^  1000,  he  had  |  of 
his  original  capital,  what  was  his  original  capital  ? 

81.  A  cistern  can  be  filled  by  one  pipe  in  20  minutes,  by 
another  in  15  minutes,  and  it  can  be  emptied  by  a  third  in  10 


SIMPLE  EQUATIONS  183 

minutes.  If  the  three  pipes  are  running  at  tlie  same  time,  how 
long  will  it  take  to  till  the  cistern  ? 

82.  A  man  walked  from  A  to  B  at  the  rate  of  2  miles  an  hour, 
and  rode  back  at  the  rate  of  3J  miles  an  hour,  being  gone  13 
hours.     How  far  is  it  from  A  to  B  ? 

83.  An  express  train  whose  rate  is  40  miles  an  hour  starts 
1  hour  and  4  minutes  after  a  freight  train  and  overtakes  it  in 
1  hour  and  36  minutes.  How  many  miles  does  the  freight  train 
run  per  hour  ? 

84.  The  distance  from  Albany  to  Syracuse  is  148  miles.  A 
canal  boat  leaves  Albany  for  Syracuse,  moving  at  the  rate  of 

3  miles  in  2  hours;  at  the  same  time  another  leaves  Syracuse  for 
Albany,  moving  at  the  rate  of  5  miles  in  4  hours.  How  far  from 
Albany  do  they  meet  ? 

85.  A  steamer  goes  5  miles  downstream  in  the  same  time  that 
it  goes  3  miles  upstream ;  but  if  its  rate  each  way  is  diminished 

4  miles  an  hour,  its  rate  downstream  will  be  twice  its  rate  up- 
stream.    How  fast  does  it  go  in  each  direction  ? 

86.  A,  B,  and  C  together  can  do  a  piece  of  work  in  ^\  days; 
B  can  do  \  as  much  as  A,  and  C  can  do  |  as  much  as  B  in  a  day. 
In  how  many  days  can  each  do  the  work  alone  ? 

87.  A  can  do  a  piece  of  work  in  G  days  that  B  can  do  in  14 
days.  A  began  the  work,  and  after  a  certain  number  of  days  B 
took  his  place  and  finished  the  work  in  10  days  from  the  time 
it  was  begun  by  A.     How  many  days  did  B  work  ? 

88.  In  a  certain  weight  of  gunpowder  the  niter  composed  10 
pounds  more  than  J  of  the  weight,  the  sulphur  3  pounds  more 
than  yV,  and  the  charcoal  3  pounds  less  than  ^^  of  the  niter. 
What  was  the  weight  of  the  gunpowder  ? 

89.  A  library  containing  16,000  volumes  was  divided  into  five 
departments.  In  the  department  of  history  there  were  twice  as 
many  volumes  as  in  the  department  of  science,  and  500  more 
than  \  as  many  volumes  as  in  the  juvenile  department.  Of 
fiction  there  were  1\  times  as  many  volumes  as  of  science,  and 
500  less  than  8  times  as  many  as  in  the  reference  department 
How  many  volumes  were  there  in  each  department  ? 


184  ACADEMIC  ALGEBRA 

90.  An  estate  was  divided  among  four  heirs,  A,  B,  C,  and  D. 
If  the  value  of  the  estate  had  been  f  1000  less,  what  A  received 
would  have  been  i  of  it,  and  what  B  received  ^  of  it ;  if  the  value 
of  the  estate  had  been  ^  greater,  what  C  received  would  have 
been  4-  of  it,  and  what  D  received  i  of  it.  What  sum  did  each 
receive  ? 

91.  A  father  takes  3  steps  while  his  son  takes  5 ;  but  2  of  the 
father's  steps  are  equal  to  3  of  the  son's.  How  many  steps  will 
the  son  require  to  overtake  his  father,  who  is  36  steps  ahead  ? 

92.  A  purse  contained  some  money  and  a  ring  worth  ^10 
more  than  the  money.  If  the  purse  was  worth  i  as  much  as  the 
money  it  contained,  and  the  purse  and  the  money  together  were 
worth  1  as  much  as  the  ring,  what  was  the  value  of  each  ? 

93.  Brass  is  8|  times  as  heavy  as  water,  and  iron  7i  times  as 
heavy  as  water:  A  mixed  mass  weighs  57  pounds,  and  when 
immersed  displaces  7  pounds  of  water.  How  many  pounds  of 
each  metal  does  the  mass  contain  ? 

94.  A  man  began  business  with  $4725,  and  annually  added  to 
his  capital  ^  of  it.  At  the  end  of  each  year  he  put  aside  a 
certain  sum  for  expenses.  If  at  the  end  of  the  third  year,  after 
taking  out  the  sum  for  expenses,  his  capital  was  f  3800,  what 
were  his  annual  expenses  ? 

95.  The  sum  of  two  numbers  is  s,  and  their  difference  d. 
What  are  the  numbers  ? 

Solution 
Let  X  =  the  greater  number. 

Then,  x  —  d  =  the  less  number, 

and  x  +  x  —  d  =  s; 

,'.  X  —    ^    ,  the  greater  number,  (1) 

'  2i 

and  X  —  d  = ,  the  less  number.  .  (2) 

If  the  sum  of  two  numbers  is  30,  and  their  difference  is  6, 
what  are  the  numbers  ? 

By  (1),  the  greater  number  is        "^    ,  or  18  ; 
by  (2),  the  less  number  is  SO -6^  ^^  ^^2, 


SIMPLE  EQUATIONS  185 

A  problem  in  which  the  numbers  assumed  to  be  known  are 
represented  by  letters  to  which  any  values  may  be  assigned  is 
called  a  General  Problem. 

Problem  95  is  a  general  problem. 

The  results  obtained  in  solving  a  general  problem  may  be  con- 
sidered formulce  for  solving  similar  problem^. 

96.  Divide  c  cents  between  two  boys  so  that  one  shall  have  d 
cents  more  than  the  other.  If  c  =  50  and  d  =  10,  how  much  will 
each  receive  ? 

97.  A  horse  and  a  saddle  are  together  worth  a  dollars,  and  the 
horse  is  worth  m  times  as  much  as  the  saddle.  What  is  the  value 
of  each  ?     What,  if  a  =  160  and  m  =  3  ? 

98.  Divide  h  into  two  parts  one  of  which  represents  m  times 
the  other.     What  will  they  be,  if  h  represents  100,  and  m,  4  ? 

99.  An  estate  of  a  dollars  is  divided  between  two  heirs  in  the 
proportion  of  m  to  n.  What  is  the  share  of  each  ?  What  is  the 
share  of  each,  if  a  =  40,000,  m  =  5,  and  w  =  3  ? 

100.  If  A  can  do  a  piece  of  work  in  a  days,  and  B  in  6  days, 
in  what  time  can  both  do  it  working  together  ?  Give  the  result, 
if  a  =  10  and  h  — 15. 

101.  An  alloy  of  two  metals  is  composed  of  m  parts  of  one  to 
n  parts  of  the  other.  How  many  pounds  of  each  are  required  in 
the  composition  of  a  pounds  of  the  alloy  ? 

Bell  metal  is  an  alloy  of  5  parts  of  tin  and  16  parts  of  copper. 
How  many  pounds  of  tin  and  of  copper  are  there  in  a  bell  which 
weighs  4200  pounds  ? 

102.  A  wheelman  set  out  from  B  at  the  rate  of  r  miles  an  hour. 
a  hours  later  another  started  in  pursuit  at  the  rate  of  p  miles  an 
hour.  How  far  from  B  will  the  second  wheelman  overtake  the 
first  ?     What  will  be  the  distance,  if  r  =  10,  p  =  12,  and  a  =  8  ? 

103.  A  man  traveled  from  home  at  the  rate  of  a  miles  an  hour 
and  returned  at  the  rate  of  h  miles  an  hour.  If  he  made  the 
entire  journey  in  li  hours,  how  far  from  home  did  he  go  ?  How 
far,  if  a  =  4,  6  =  3i,  and  /i  =  15? 


SIMULTANEOUS    SIMPLE    EQUATIONS 


TWO  UNKNOW^N  NUMBERS 

201.  1.  If  £c  +  2/ =  12,  what  is  the  value  of  x?  of?/?  How 
many  values  may  x  have  ?     How  many  may  y  have  ? 

2.  In  the  expression  x  -\-  y  =  12,  x  and  y  each  may  have  an 
indefinite  number  of  values,  but  if,  at  the  same  time,  x  —  y  =  4:, 
what  is  the  value  of  x?  of  y? 

3.  Although  one  equation  containing  two  unknown  numbers 
has  an  indefinite  number  of  values  for  each  unknown  number,  or 
is  indeterminate,  what  can  be  said  about  the  values  of  the  un- 
known numbers,  when  two  equations  are  given  involving  the  same 
values  of  the  unknown  numbers,  but  in  different  relations,  that  is, 
when  two  independent  equations  are  given  ? 

^202.    Two  or  more  equations  in  which  the  unknown  numbers 
have  the  same  values  are  called  Simultaneous  Equations. 

If  X  and  y  represent  the  same  numbers  in2a;  +  3?/  =  19as  they  represent 
in  5  X  —  ?/  =  22,  2  a:  +  3  y  =  19  and  5  cc  —  ?/  =  22  are  simultaneous  equations. 

203.  Equations  that  represent  different  relations  between  the 
unknown  numbers,  and  so  cannot  be  reduced  to  the  same  form,  are 
called  Independent  Equations. 

3  X  +  3  2/  =  18  and  2  x  +  2  ?/  =  12  really  express  but  one  relation  between 
X  and  y  ;  viz.,  that  their  sum  is  6.  Hence,  both  equations  may  be  reduced  to 
the  same  form,  as  x-\-y  =  Q.  But  3  x  +  3  ?/  =  18  and  x  +  3  ?/  =  14  express 
different  relations  between  x  and  y^  and  cannot  be  reduced  to  the  same  form. 
Hence,  they  are  independent  equations. 

204.  An  equation  whose  unknown  numbers  may  have  an  in- 
finite number  of  values  is  called  an  Indeterminate  Equation. 

X  -f  y  =  6  is  an  indeterminate  equation,  because,  if  x  =  2,  «/  =  4  ;  if  x  =  3, 
2/  =  3 ;  \ix  =  ^,y  =  ^^,  etc. 

186 


SIMULTANEOUS   SIMPLE   EQUATIONS  187 

205.  Principle.  —  Every  single  equation  involving  two  or  more 
unknown  numbers  is  indeterminate. 

206.  The  process  of  deriving  from  a  set  of  simultaneous  equa- 
tions, equations  involving  a  less  number  of  unknown  numbers 
than  is  found  in  the  given  equations  is  called  Elimination. 

207.  Two  sets  of  simultaneous  equations  each  having  all  the 
roots  of  the  other  set  are  called  Equivalent  Systems  of  equations. 

{s.WrJn}    and    {11X11  =  11} 

are  equivalent  systems,  for  each  system  is  satisfied  by  the  same  set  of  values, 
X  =  5  and  ?/  =  3,  and  neither  is  satisfied  by  any  other  set  of  values. 

208.  Any  equation  in  a  set  of  simultaneous  equations  may  be 
transformed  into  an  equivalent  equation  by  employing  the  prin- 
ciples of  equivalency  stated  in  §  196,  since  these  principles  apply 
to  all  equations.  Hence,  it  remains  to  seek  the  principle  by  which 
simultaneous  equations  are  combined  in  the  process  of  elimination 
without  introducing  or  losing  roots. 

This  Principle  of  Elimination  may  be  stated  as  follows,  a  and  b 
being  known  multipliers,  not  zero,  and  either  positive  or  negative  : 

If  any  equation  of  a  system  is  replaced  by  the  s^im  or  difference 
of  a  times  that  equation  and  b  times  another  equation  of  the  system^ 
the  resulting  system  is  equivalent  to  the  given  system. 

The  proof  is  as  follows : 

By  §  190,  Prin.  1,  all  the  terms  of  the  second  member  of  an  equation  may 
be  transposed  to  the  first  member. 

Then,  let  ^  =  0  i 

^  =  0  (1) 


be  the  given  system,  and  let  a  and  h  be  any  known  multipliers  except  zero 
and  either  positive  or  negative. 
It  is  to  be  proved  that  the  system 


aA+hB  =  0 
B  =  0 


(2) 


is  equivalent  to  the  given  system  (1). 

Since  a  and  b  are  known  multipliers,  not  zero,  by  §  196,  Prin,  2,  every  set 
of  values  of  the  unknown  numbers  that  makes  ^  =  0  makes  aA  =  0,  and 
every  set  of  values  that  makes  J5  =  0  makes  bB  =  0.     Hence,  every  set  of 


188  ACADEMIC  ALGEBRA 

values  that  satisfies  (1)  makes  a^  +  hB  equal  to  zero  and  thus  satisfies  (2), 
since  all  of  the  equations  of  (1)  and  (2)  except  the  first  are  the  same. 

Again,  since  every  set  of  values  that  satisfies  (2)  makes  J5,  or  hB  equal  to 
zero  and  also  makes  aA-\-hB  =  0,  every  such  set  of  values  makes  aA  =  0 
and  therefore,  by  §  196,  Prin.  2,  makes  J.  =  0  and  satisfies  (1). 

Since  all  the  roots  of  (1)  are  roots  of  (2)  and  all  the  roots  of  (2)  are  roots 
of  (1),  (1)  and  (2)  are  equivalent  systems. 

209.    Elimination  by  Addition  and  Subtraction. 

1.  If  2  ic  -f-  2  2/  =  10  and  3  a?  —  2  ?/  =  5  are  added,  what  is  the 
resulting  equation?  When  may  an  unknown  number  be  elimi- 
nated by  addition  f 

2.  Ifaj  +  22/  =  5is  subtracted  from  ^x-\-2y  =  11,  what  is 
the  resulting  equation  ?  When  may  a  number  be  eliminated  by 
subtraction  9 

Principle.  —  A  literal  number  having  the  same  coefficient  in 
two  equatioyis  may  be  eliminated  by  adding  the  equations,  if  the 
coefficients  have  unlike  signs,  or  by  subtracting  one  equation  from  the 
other,  if  the  coefficients  have  like  signs. 

*  Examples 

1.   Find  the  value  of  x  and  of  2/  in  2  x-{-S  y  =  1  and  3  a;-f-4  ^=10. 

Explanation.  —  Since  x  has  not  the  same  co- 
efficients in  both  equations,  the  equations  are 
multiplied  by  such  numbers  as  will  make  the  co- 
efficients of  X  alike.  Multiplying  eq.  (1)  by  3  and 
eq.  (2)  by  2  gives  equations  (3)  and  (4).  From 
these  X  is  eliminated  by  subtraction,  and  the  value 
of  y  is  found. 

The  value  of  x  may  be  found  by  multiplying 
eq.  (1)  and  (2)  by  such  numbers  as  will  make 
the  coefficients  of  y  equal  and  then  subtracting 
the  resulting  equations. 

Or,  the  value  of  x  may  be  found  by  substituting 
the  value  of  y  for  y  in  one  of  the  given  equations,  as  eq.  (1). 

Proof  op  the  Equivalence.  —By  the  principle  of  elimination,  the  system 
(1,5),  obtained  by  subtracting  (4)  from  (3),  is  equivalent  to  the  given  system 
(1,  2).     Hence,  the  only  value  of  y  in  the  given  system  is  1. 

Since  y  represents  1  in  the  given  system,  (1)  is  equivalent  to  (6),  which 
by  §  196  is  equivalent  to  (7).  Hence,  the  given  system,  being  equivalent  to 
(1,  5),  is  equivalent  to  (7,  6),  which  is  satisfied  by  one  and  only  one  set  o'i 
values,  X  =  2  and  y  =  1. 


PROCESS 

2x  +  Zy=    7 
3  ic  +  4  ?/  =  10 

(1) 

(2) 

6  a;  +  9  1/  =  21 
6  x  +  8 1/  =  20 

(3) 
(4) 

y=   1 

2«+3=    7 

x=    2 

(5) 
(6) 
(7) 

SIMULTANEOUS  SIMPLE  EQUATIONS 


189 


Rule.  —  If  necessary,  multiply  the  equations  by  such  numbers  as 
ivill  cause  the  coefficients  of  one  letter  to  be  numerically  equal  in  the 
resulting  equations. 

When  the  sigyis  of  these  coefficients  are  unlike,  add  the  equations; 
when  the  signs  are  alike,  subtract  one  equation  from  the  other. 


Solve  by  addition  or  subtraction : 

7  ic  —  5  V  =  52, 
2.    '  J  y 


2  ic  4-  5  ?/  =  47. 

[  3  «  4-  2  ?/  =  23, 
.  a;  +  y  =  8. 

3  a;  -  4  2/  =  7, 
2^. 


(6x  —  ^y  = 


rzx 
*'   \2x 


2  a;  -  10  ?/  =  15, 
47/  =  18. 


6. 


9. 


10 


11 


12 


f  3  a;  -  ?/  =  4, 
\x+3y  =  -2. 

4  a;  -  2/  =  19, 
a; +  3  2/ =  21. 

^x  +  2y  =  6, 
\2x  +  y  =  l.  ^ 

2  a; +  3  2/ =  17, 

[  3  a;  +  2  2/  =  18. 

I  3  a;  +  4  2/  =  25, 
I  4  a? +  3  2/ =  31. 

5  a;  +  6  2/  =  32, 
7  a;  -  3  y  =  22. 

rSx  +  6y  =  39, 
1  9  a; -41/ =  51. 


13 


14 


15 


(7x-9y  =  6, 
'    [x  +  2y  =  U. 

rl3i 

l4a; 

fSx-3y  =  U, 
\7x-5y 


lSx-y  =  20, 
+  2y  =  20. 


16. 


17. 


29. 

6  a;  -  5  2/  =  33, 

4  a;  4-  4  2/  =  44. 

f  X  -h  14  2/  =  38, 
14  a;  +  2/  =  142. 

5  a;  4-  2/  =  12, 


1  a; +  5  2/ =  36. 

I 


19. 


20. 


3.^4- 11 2/ =  67, 
5  a;  —  3  2/  =  5. 


21.   ^ 


?  +  ^  =  12, 

4     2 

^-2^  =  -2. 
4      2 


3^3"'' 


^4-^ 


6i 


190  ACADEMIC  ALGEBRA 

210.   Elimination  by  Comparison. 

1.  If,  in  the  simultaneous  equations  x  —  y  =  ^  and  3*  +  4  ?/  =13, 
the  terms  containing  y  are  transposed,  what  will  be  the  resulting 
equations  ? 

2.  Since  the  second  members  of  these  derived  equations  are 
each  equal  to  ic,  how  do  they  compare  with  each  other  ? 

3.  If  these  second  members,  then,  are  placed  equal  to  each 
other,  how  many  unknown  numbers  will  this  equation  contain  ? 

4.  How  may  an  unknown  number  be  eliminated  from  two 
simple  simultaneous  equations  by  comparison^ 

Examples 
1.    Find  the  value  of  a;  and  of  y  in  2ic— 3?/=10  and  5;c+2?/=6. 

Explanation.  —  Solving  eq.  (1)  as  if 
X  were  the  only  unknown  number,  the 
value  of  X  is  given  as  in  eq.  (3).  In  like 
manner,  solving  eq.  (2)  for  x,  another 
expression  is  found  for  the  value  of  ic,  as 
in  eq.  (4). 

Since  the  equations  are  simultaneous, 
the  unknown  numbers  have  the  same 
values  in  each,  and  the  two  values  of  x 
form  an  equation,  as  eq.  (5),  from  which 
X  has  been  eliminated.  Solving  eq.  (5), 
the  value  of  y  is  found  to  be  —  2,  eq.  (6). 

By  a  similar  process  the  value  of  x  may 
be  found  ;  or,  substituting  the  value  of  y 
for  y  in  one  of  the  preceding  equations, 
as  (3),  the  value  of  x  is  found  to  be  2, 
eq.  (7). 

Proof  of  the  Equivalence.  —By  §  196,  (3)  is  equivalent  to  (1)  and  (4) 
is  equivalent  to  (2).  Hence,  the  system  (3,  4)  is  equivalent  to  the  given 
system. 

Since  (5),  which  by  §  196  is  equivalent  to  (6),  may  be  obtained  by  sub- 
tracting (3)  from  (4)  and  transposing,  by  the  principle  of  elimination,  §  208, 
(6,  1),  or  (6,  3),  is  equivalent  to  (4,  3),  or  to  (2,  1),  the  given  system. 

But  since  —  2  is  the  only  value  of  y  in  the  given  system,  by  §  196,  (7)  is 
equivalent  to  (3),  or  to  (1).  Hence,  (7,  6),  which  is  satisfied  by  one  and 
only  one  set  of  values,  x  =  2  and  ?/  =  —  2,  is  equivalent  to  the  given  system. 
Therefore,  the  given  system  is  satisfied  by  x  =  2,  y  =  —  2  and  by  no  other 
values  of  x  and  y. 


PROCESS 

2a: -32/ =  10 

(1) 

5  a; +  2?/=    6 

(2) 

10  +  3  y 

(3) 

o 

(4) 

10 +  31!/      0-2;/ 
2                5 

(5) 

■  ••  2/  =  -2 

(6) 

.  =  10-*' 

2 

(3) 

x  =  2 

(') 

SIMULTANEOUS  SIMPLE  EQUATIONS 


191 


Rule.  —  Find  an  expression  for  the  value  of  the  same  unknown 
number  in  each  equation,  equate  the  two  expressions,  and  solve  the 
equation  thus  formed. 


Solve  by  comparison  : 
2. 


3. 


4. 


5. 


7. 


'3a; -2?/ =  10, 
a;  4-  2/  =  70. 


^x  +  y  =  22, 
x  -|-  5 1/  =  14. 

2a; +  32/ =  24, 
5a;-3y=18. 

3  a;  +  5  ?/  =  14, 
2a;-3?/  =  3. 

r3a;4-2?/  =  36, 
|5a;-92/  =  23. 

|2a;  +  77/  =  8, 
|3a;  +  9y  =  9. 


r4a;-f  6?/  =  l< 
\Sx-2y=l 


9. 


10. 


11. 


12. 


13. 


14. 


15. 


4a;  +  32/  =  34, 
11  a; +  52/ =  87. 

r4.T-132/  =  5, 
I  3  a; +11 2/ =-17. 

rl8a;-32/  =  42/, 
|l_4a;  +  32/  =  27. 

|72/-»  =  0, 
1  a; +  22/ =  18. 

r  32/ +  9  =  5a;, 
|l6-2a;  =  52/. 

f5a;-40  =  2/, 
I  5?/ -60  =  a;. 

r22/-lla;  =  67, 


l2 


a;  +  5y  =  20. 


211.   Elimination  by  Substitution. 

1.  In  an  equation  containing  two  unknown  numbers,  if  the 
value  of' one  is  found,  how  is  the  value  of  the  other  obtained? 

2.  Express  the  value  of  x  in  the  first  of  the  simultaneous 
equations  a;  +  2/  =  5  and  x  -\-2y  =1  by  transposing  y  to  the 
second  member. 

3.  When  this  expression  for  the  value  of  x  is  substituted  for 
X  in  the  second  equation,  how  many  unknown  numbers  does  the 
resulting  equation  contain  ? 

4.  How  may  an  unknown  number  be  eliminated  from  simul- 
taneous equations  by  substitution'^ 


192 


ACADEMIC  ALGEBRA 


PROCESS 

x+5y=9 

(1) 

Sx-2y  =  10 

(2) 

x  =  9-5y 

(3) 

3(9-5y)-2y  =  10 

W 

.-.  2/  =  l 

® 

x  =  9- 

■5    (3) 

X=4: 

(6) 

Examples 

1.   Find  the  value  of  x  and  of  y  in  x-\-5y=9  and  3  a;— 22/=10. 

Explanation. — Solving  eq.  (1) 
for  X,  X  =  9  —  L)y. 

Since  the  given  equations  are 
simultaneous,  x  has  the  same  value 
in  eq.  (2)  as  in  eq.  (1). 

If  9  —  5y  is  substituted  for  x  in 
eq.  (2),  the  resulting  equation  will 
be  true.  Substituting  and  solving, 
y  =  h 

Substituting  the  value  of  y  for  y 
ineq.  (3),  a:  =  4,  eq.  (G). 
Proof  of  the  Equivalence.  — By  §  196,  (3)  is  equivalent  to  (1).  In  the 
process,  (4)  is  obtained  by  substituting  the  expression  equal  to  x  in  (3)  for  a; 
in  (2),  or  by  substituting  3(9  —  5  ?/)  for  3  a;  in  (2).  Since  the  substitution 
oi  3(9  -  by)  for  3 x  may  be  performed  by  subtracting  3 x  =  3(9  —  5  ?/)  from 
(2),  by  the  principle  of  elimination  (4),  or  the  equivalent  equation  (5),  may 
take  the  place  of  (2)  in  the  system  (3,  2)  equivalent  to  the  given  system. 
Hence,  (3,  5)  is  equivalent  to  (1,  2). 

Since  the  only  value  of  y  in  the  system  (3,  5)  is  1,  (6)  is  equivalent  to  (3); 
and  (6,  5),  which  is  satisfied  by  only  one  set  of  values,  a;  =  4,  y  =  1,  is  equiva- 
lent to  (3,  5)  and  therefore  to  (1,  2). 

Rule.  —  Find  an  expression  for  the  value  of  one  of  the  unknown 
numbers  in  one  of  the  equations. 

Substitute  this  value  for  that  unknown  number  in  the  other  equa- 
tion, and  solve  the  resultiyig  equation. 


Solve  by  substitution : 

2.     J»-2'  =  4, 
[  4  ?/  —  a;  =  14. 

3      [x  +  y  =  10, 
\(yx  -ly=.M. 

3  a;  -  4  ?/  =  2G, 
x-Sy  =  22. 

62/ -10a;  =  14, 

y  —  X  =  ^. 

r2/4-l=3«, 
5a;-^9  =  3j/. 


7. 


8. 


10. 


11. 


r  17=  3a; +  2, 
\l  =  Zz-2x. 

|42/  =  10-a;, 
iy-x  =  5.  ,/ 

7z-3a;  =  18, 


2z-i>x  =  l. 

(^-Wy  =  -X, 
j  3 +  15?/  =  4a;. 

n-x  =  ^y, 

I  3(1 -a;)  =  40 -2/. 


SIMULTANEOUS    SIMPLE   EQUATIONS 


193 


Solve  by  any  method,  eliminating  without  clearing  of  fractions, 
when  possible : 


12.   \ 


^  +  1  =  11, 


19.    { 


x-\-y     x  —  y 


8, 


3  4 


13. 


^  +  ^  =  21, 
4        5 

2^  +  «i'  =  17. 


20. 


2     3  ' 

2a!-l      3y-1^5 
2  3  6* 


14.   ■! 


^  =  11-^, 
3  2 

3^  7 


X. 


21.  .^ 


5v— 7  ,  4a:— 3      io     - 


15. 


-|-42/=15, 


2 
^6'^~3 


^•^.V  =  6, 


/ 


22. 


1-12  =  1  +  8, 

^  +  1  =  2^^  +  35. 


16. 


[x-1 
4 


+  42/  =  9. 


r    1 


3 


23. 


a;  —  1      x-\-y 
3 


=  0, 


x  —  y 


+  3  =  0 


17.   \ 


[3^      2^_^^ 
4  +  3  "*^  ' 

aj  ,  3  v     ^r- 

2  +  T  =  ''- 


24. 


?-12  = 


_y  +  32 


^  4.  3^-^y  _  2r> 


18. 


3      2' 
[3      3 

ACAD.    ALG. 13 


25. 


.2  y  +  .5  ^  .49  a;  -  .7 

1.5  4.2      ' 

.5  a; -.2  ^41      1.5y-ll 

1.6  16  8 


194 


ACADEMIC  ALGEBRA 


26. 


^x      y  ,   -lo 


27. 


28. 


f  a;  +  i(3a^  -  2/  -  1)  =  1  +  I  0/  -  1), 

~4~'  +  T^^6"^^^  +  "^~"' 

8y  +  7      6a;-3y^  .      4?/-9 
10      "^2(2/ -4)         "^       5 


29. 


30. 


\Zx 

-5 

.V     2x 

-8.V 

-9 

31 

3 

12 

"12 

l(? 

n^ 

-')- 

/4ic- 

y 

8 

-)= 

5 
"6 

iC- 

-20 

22/- 
23- 

a;  — 

2 

59 

—  ? 

^0^-18  3 


31. 


3a;+6       a;  +  5  ^6a;-2 
7  3?/-5~     .14     ' 

6       "^5a?-7  9 


32. 


2  +  ~T~-^^  +  ^o^::^' 

2y-3      83-8y^^Q_ 
[  2/  +  8  ^       8  ^ 


33. 


22/- 


4a;  H- 
50- 


17 -3  a; 

y-1 

5  a; -10 


X 

2       ^      16a; 


=  82/  + 


147  -  24  2, 


SIMULTANEOUS   SIMPLE   EQUATIONS 


195 


34.    Solve 


(2)  X  2, 
(3)-(l), 


14 


X      y 


Substituting  the  value  of 


25 

3* 

Solution 

4_  3  ^14 
X      ?/       5 

2_^  5  _25 
X       y        3 

4_^10_50 
JC      y        3 

13      208 
y       16 

1_16 
y      15 

^      16 

y 

in  equation  (1), 

4      48_14 
X      15       5 

1  _3 
X     2 

(1) 

(2) 
(3) 
(4) 
(6) 
(6) 

(7) 
(8) 
(9) 


In  fractional  equations  in  which  the  denominators  are  simple  expressions 
like  the  above,  an  unknown  number  should  be  eliminated  before  clearing  of 
fractions. 


35. 


36. 


?/_ 


2, 


2       3 


12. 


13, 


4?/  _^__o 
3       3~        * 


37. 


38.     \ 


^  +  ^  =  64, 
X      y 

5  +  ?  =  73i. 
X      y 

10  +  ^  =  20, 

X       y 

X       y 


196 


ACADEMIC  ALGEBRA 


39. 


40. 


41. 


5_ 

3_ 

_ 

X 

y 

25 

X 

y 

=  6. 

2 

X 

3^ 

y 

=  5, 

5_ 

X 

2_ 

y 

:7. 

X      y 

9 

8' 

X      y 

11 
12' 

2, 


42. 


43. 


^^  +  ^-  =  30, 
X      y 

-  +  ?  =  30. 
2/      a; 


y 

2a;      y 


23. 


44.     \ 


(   7         2 

- ^  =  10 

ISx     3y         ' 


5         2 
6a;      11  1/ 


17. 


LITERA.L  Simultaneous  Equations 


1.    Solve 


(l)xd, 
(2)x6, 
(3) -(4), 


(l)xc, 
(2)  X  a, 
(7) -(6), 


aa;  +  by 

=  m, 

ex  +  dy 

=  n. 

Solution 

ax  +  by  —  m 

cx  +  dy  =  n 

adx  +  bdy  =  dm 

hex  +  bdy  =  bn 

(ad  -  bc)x  =  dm  - 

-6w 

.;x  =  ^'^- 
ad- 

-6c 

acx  +  bey  =  cm 

acx  +  ady  =  an 

(ad  —  bc)y  =  an  - 

cm 

.   «  -  «w  - 

-  cm 

ad  —  &c 


(1) 

(2) 
(3) 

(4) 

(5) 

(6) 
(7) 

(8) 


In  literal  simultaneous  equations,  elimination  is  usually  performed  by  the, 
method  of  addition  and  subtraction. 


SIMULTANEOUS  SIMPLE  EQUATIONS 


197 


7. 


ax+  by  =  m, 
bx  —  ay=  c. 

CLx  —  by  =  m, 
.  ex  —  dy  =  r. 

ax  =  by, 
X  -\-  y  =  ab. 

X  —ay  =  71, 
bx-\-y=p. 

a(x-y)  =  5, 
bx  —  cy=:  n. 

a(a-x)  =  b(y-b), 
ax  =  by. 

x-{-y=:b-a, 

bx  —  ay  +  2  ab  =  0. 

X  y  a 
1_1^1 
X     y     b 


a 


10. 


=  -1, 


11. 


12. 


6_a__^ 


a     b 


l  ab     ab 


a     b 

bx  —  ay  =  0. 


13. 


14. 


15. 


16. 


17. 


a     b 
^_^_1 
6      a~2 


1    ,    1 

— +  — =  c, 

ax     by 

=  d 


a  ,  b 
a;     y 

x      y 

2/  +  l~a-6  +  l' 
a;  —  y  =  2  6. 


o-j-y 


2/. 


^  —  y^y  —  (^. 

a  b 


18. 


19. 


20. 


a;  —  a     a  —  2/ 

^±y=a. 

x-y 


a     0 
6      c 


a      ,      & 


a  +  a;     &  —  2/     ^ 
b  a     _  b^ 

^  a-\^x     b  —  y     a 


198  ACADEMIC  ALGEBRA 

Problems 

1.  There  are  two  numbers  such  that  if  twice  the  first  is  added 
to  3  times  the  second,  the  sum  wall  be  130 ;  but  if  5  times  the 
first  is  diminished  by  the  second,  the  remainder  will  be  70. 
What  are  the  numbers  ? 


Solution 

Let 

X  =  the  first  number, 

and 

y  =  the  second  number. 

Then, 

2x  +  Sy  =  130, 

and 

5  X  -    y  =  70. 

Eliminating  y, 

nx  =  340, 

a;  =  20. 

Whence, 

by  substitution, 

>--B0. 

2.  A  drover  sold  3  cows  and  7  horses  to  one  person  for  $  600, 
and  to  another  person,  at  the  same  prices,  3  cows  and  3  horses  for 
$  300.     How  much  per  head  did  he  get  for  each  ? 

3.  With  $  30  a  man  can  buy  20  yards  of  one  kind  of  cloth  and 
50  yards  of  another ;  with  $  23  he  can  buy  30  yards  of  the  first 
kind  and  20  yards  of  the  second  kind.  W^hat  is  the  price  of  each 
per  yard  ? 

4.  If  45  bushels  of  wheat  and  37  bushels  of  rye  together  cost 
$  62.70,  and  37  bushels  of  wheat  and  25  bushels  of  rye,  at  the 
same  prices,  cost  $  48.30,  what  is  the  price  of  each  per  bushel  ? 

5.  Henry  expended  95  cents  for  apples  and  oranges,  paying 
5  cents  for  each  orange  and  4  cents  for  each  apple.  If  he  had  22 
of  both,  how  many  of  each  did  he  buy  ? 

6.  Five  years  ago  A  was  J  as  old  as  B,  and  10  years  hence  he 
will  be  ^  as  old  as  B.     What  are  their  ages  ? 

7.  A  said  to  B,  "  If  you  were  twice  as  old,  and  I  were  ^  as  old, 
or  if  you  were  ^  as  old,  and  I  were  3  times  as  old,  the  sum  of 
our  ages  would  be  70."     How  old  was  each  ? 

8.  A  boy  is  given  28  cents  to  buy  a  dozen  cakes.  He  finds 
that  some  cost  2  cents  each  and  some  3  cents  each.  How  many 
of  each  kind  can  he  purchase  ? 


SIMULTANEOUS   SIMPLE   EQUATIONS  199 

9.  A  said  to  B,  "Give  me  $20, and  I  shall  have  3  times  as 
much  money  as  you."  B  replied,  "  Give  me  $  5,  and  I  shall  have 
twice  as  much  money  as  you."     How  much  money  had  each  ? 

Solution 

Let  X  =  the  number  of  dollars  A  had, 

and  y  =  the  number  of  dollars  B  had. 

Then,  a;  +  20  =  3(?/ -  20),  * 

and  y+    5  =  2(x  -    5). 

Solving,  X  —  25,  the  number  of  dollars  A  had, 

and  y  =  35,  the  number  of  dollars  B  had. 

10.  If  A  gives  B  $  100,  B  will  have  4  times  as  much  money  as 
A ;  but  if  B  gives  A  $  200,  A  will  have  4  times  as  much  money 
as  B.     What  sum  of  money  has  each  ? 

11.  A  said  to  B,  "Give  me  20  cents  of  your  money,  and  I  shall 
have  half  as  much  as  you."  B  replied,  "Give  me  25  cents  of 
your  money,  and  I  shall  have  5  times  as  much  as  you."  How 
much  had  each  ? 

12.  If  A  had  $300  more,  he  would  have  twice  as  much  as  B; 
if  B  had  $  300  less,  he  would  have  \  as  much  as  A.  How  much 
money  has  each  ? 

13.  If  1  is  added  to  each  term  of  a  fraction,  its  value  will 
be  I ;  if  1  is  subtracted  from  each  term  of  the  fraction,  its  value 
will  be  ^.     What  is  the  fraction  ? 

Solution 

Let  -  represent  the  fraction. 

y 

and  ^  =  h 

y-i     2 

Solving,  a;  =  3, 

and  y  =  5. 

Q 

That  is,  -  is  the  fraction. 

6 


4 

200  ACADEMIC  ALGEBRA 

14.  If  1  is  added  to  the  numerator  of  a  certain  fraction,  its 
value  becomes  f ;  if  2  is  added  to  the  denominator,  its  value 
becomes  |.     What  is  the  fraction  ? 

15.  Find  a  fraction  that  is  equal  to  ^  when  its  terms  are 
diminished  by  2,  and  is  equal  to  f  when  its  terms  are  increased 
by  2. 

16.  A  certain  number  expressed  by  two  digits  is  equal  to  7 
times  the  sum  of  its  digits ;  if  27  is  subtracted  from  the  number, 
the  difference  will  be  expressed  by  reversing  the  order  of  the 
digits.     What  is  the  number? 


Solution 

Let 

X  =  the  digit  in  tens'  place. 

and 

y  z=  the  digit  in  units'  place. 

Then, 

10x  +  y  =  t'he  number. 

and 

10  y  +  ic  =  the  number  with  its  digits  reversed ; 

.-.  10x-{-y  =  7ix  +  y), 

and 

lOx  +  y  -21  =  10y  -{-x. 

Solving, 

x  =  6, 

and 

y  =  s. 

Hence, 

10  a;  +  y  =  60  +  3,  or  63,  the  number. 

17.  The  sum  of  the  two  digits  of  a  certain  number  is  12,  and 
the  number  is  3  greater  than  6  times  the  sum  of  its  digits. 
What  is  the  number? 

18.  When  a  certain  number  expressed  by  two  digits  is  divided 
by  the  sum  of  its  digits,  the  quotient  is  8  ;  and  when  the  first 
digit  is  diminished  by  3  times  the  second,  the  remainder  is  1. 
What  is  the  number? 

19.  The  sum  of  the  two  digits  of  a  number  is  12.  If  the 
order  of  the  digits  is  reversed,  the  number  will  lack  12  of  being 
doubled.     What  is  the  number  ? 

20.  A  farmer  bought  100  acres  of  land  for  $  3250.  If  part  of 
it  cost  him  $  40  an  acre  and  the  rest  of  it  ^  15  an  acre,  how  many 
acres  were  there  of  each  kind  ? 


SIMULTANEOUS  SIMPLE  EQUATIONS  201 

21.  The  admission  to  an  entertainment  was  50  cents  for  adults 
and  35  cents  for  children.  If  the  proceeds  from  100  tickets 
amounted  to  $  39.50,  how  many  tickets  of  each  kind  were  sold  ? 

22.  A  man  paid  a  bill  of  $  16  in  25-cent  pieces  and  5-cent 
pieces.  If  the  number  of  coins  was  80,  how  many  of  each  kind 
were  there  ? 

23.  A  man  paid  $  1  for  some  apples  at  3  cents  each  and  some 
oranges  at  5  cents  each.  He  sold  -^  of  the  apples  and  \  of  the 
oranges  at  cost  for  34  cents.     How  many  of  each  did  he  buy  ? 

24.  A  and  B  together  can  do  a  piece  of  work  in  12  days. 
After  A  has  worked  alone  for  5  days,  B  finishes  the  work  in  26 
days.     In  what  time  can  each  alone  do  the  work  ? 

25.  A  blacksmith  and  his  son  had  a  contract  to  make  a  certain 
number  of  horseshoes.  If  both  had  worked  together,  they  could 
have  done  the  work  in  6  days.  But  the  father  worked  8  days, 
and  the  son  finished  the  work  in  3  days.  In  how  many  days 
could  each  have  done  the  work? 

26.  A  man  and  his  two  sons  can  dig  a  ditch  in  6  days ;  if  the 
man  and  either  son  work  7  days,  the  other  son  can  complete  the 
ditch  by  working  2  days.  In  what  time  can  each  alone  dig 
the  ditch? 

27.  A  certain  number  of  persons  agree  to  share  equally  the 
expense  of  hiring  a  coach.  If  each  paid  75  cents,  there  would  be 
$  1.25  over ;  but  if  each  paid  50  cents,  there  would  be  $  2.50  lack- 
ing. What  is  the  number  of  persons  and  the  expense  of  hiring 
the  coach  ? 

28.  A  train  ran  a  certain  distance  at  a  uniform  rate.  Had  the 
rate  been  increased  5  miles  an  hour,  the  journey  would  have  been 
2  hours  shorter;  but  had  the  rate  been  diminished  5  miles  an 
hour,  the  journey  would  have  been  2\  hours  longer.  What  was 
the  distance  and  the  rate  of  the  train  ? 

\i     Suggestion.  —  Let  x  miles  per  hour  be  the  actual  rate  of  the  train  and  y 
the  number  of  hours  required  to  complete  the  journey. 

29.  A  sum  of  money  was  divided  equally  among  a  certain 
number  of  x3ersons.  If  there  had  been  4  persons  more,  the 
share  of  each  would  have  been  $3  less;  but  if  there  had  been 


202  ACADEMIC  ALGEBRA 

2  persons  less,  the  share  of  each  would  have  been  $2  more. 
Among  how  many  persons  was  the  money  divided  and  what  was 
the  share  of  each  ? 

30.  A  dealer  bad  eggs  to  sell  and  wished  to  buy  potatoes. 
He  found  that  6  dozen  eggs  were  worth  10  cents  more  than  2 
bushels  of  potatoes ;  and  that  10  dozen  eggs  were  worth  10  cents 
less  than  4  bushels  of  potatoes.  How  much  were  eggs  and 
potatoes  worth  ? 

31.  If  a  rectangular  floor  were  2  feet  wider  and  5  feet  longer, 
its  area  would  be  140  square  feet,  greater ;  if  it  were  7  feet  wider 
and  10  feet  longer,  its  area  would  be  390  square  feet  greater. 
What  are  its  dimensions  ? 

32.  If  54  is  added  to  a  certain  number,  expressed  by  two 
digits  whose  sum  is  8,  the  order  of  the  digits  will  be  reversed. 
What  is  the  number  ? 

33.  If  13  is  added  to  a  certain  number,  the  sum  will  be  equal 
to  5  times  the  sum  of  the  two  digits  of  the  number ;  and  if  36  is 
added  to  the  number,  the  order  of  its  digits  will  be  reversed. 
What  is  the  number  ? 

34.  A  and  B  can  do  a  piece  of  work  in  a  days,  or  if  A  works 
m  days  alone,  B  can  finish  the  work  by  working  n  days.  In  how 
many  days  can  each  do  the  work  ? 

35.  A  can  build  a  wall  in  c  days,  and  B  can  build  it  in  d  days. 
How  many  days  must  each  work  so  that,  after  A  has  done  a  part 
of  the  work,  B  can  take  his  place  and  finish  the  wall  in  a  days 
f  roiQ  the  time  A  began  ? 

36.  One  cask  contains  a  mixture  of  20  gallons  of  wine  and 
30  gallons  of  water,  and  another  contains  a  mixture  of  12  gallons 
of  wine  and  15  gallons  of  water.  How  many  gallons  mnst  be 
drawn  from  each  cask  to  form  a  mixture  that  will  contain  8  gallons 
of  wine  and  11  gallons  of  water  ? 

Suggestion.  —  If  x  gallons  of  the  mixture  are  drawn  from  the  first  cask, 
I X  gallons  of  it  will  be  wine. 

If  y  gallons  of  the  mixture  are  drawn  from  the  second  cask,  |  y  gallons  o*" 
it  will  be  wine. 


SIMULTANEOUS  SIMPLE  EQUATIONS  203 

37.  "If  I  had  received  3  oranges  more  for  my  money,"  said 
A,  "  they  would  have  cost  me  1  cent  less  each ;  but  if  I  had 
received  2  less,  they  would  have  cost  me  1  cent  more  each." 
How  many  oranges  had  he  bought,  and  at  what  price  each  ? 

38.  A  merchant  mixes  two  kinds  of  tea.  If  he  mixes  it  in 
parts  proportional  to  7  and  5,  the  value  of  the  mixture  is  46  cents 
a  pound.  If  he  mixes  it  in  parts  proportional  to  5  and  1,  the 
value  of  the  mixture  is  50  cents  a  pound.  What  is  each  kind  of 
tea  worth  per  pound  ? 

39.  A  man  invested  $4000,  a  part  at  5%  and  the  rest  at  4%. 
If  the  annual  income  from  both  investments  was  $175,  what  was 
the  amount  of  each  investment  ? 

40.  A  man  invested  a  dollars,  a  part  at  r%  and  the  rest  at  .s% 
yearly.  If  the  annual  income  from  both  investments  was  b 
dollars,  what  was  the  amount  of  each  investment  ? 

41.  A  sum  of  money,  at  simple  interest,  amounted  to  h  dollars 
in  t  years,  and  to  a  dollars  in  s  years.  What  was  the  principal, 
and  what  was  the  rate  of  interest  ? 

42.  A  had  a  certain  sura  invested  at  a  certain  rate  per  cent, 
and  B  had  $  100  less  invested  at  a  rate  2  %  higher.  B's  annual 
income  was  $  oQ^  greater  than  A's ;  but  if  B's  rate  upon  his  invest- 
ment had  been  only  1%  higher  than  A's  his  annual  income  would 
have  been  only  $  25  greater  than  A's.  How  much  was  invested 
by  each  man,  and  at  what  rate  ? 

43.  A  crew  can  row  8  miles  downstream  and  back,  or  12  miles 
"downstream  and  half  the  way  back  in  li  hours.  What  is  the 
rate  of  rowing  in  still  water  and  the  velocity  of  the  stream  ? 

44.  A  man  rows  15  miles  downstream  and  back  in  11  hours. 
The  current  is  such  that  he  can  row  8  miles  downstream  in  the 
same  time  as  3  miles  upstream.  What  is  his  rate  of  rowing  in 
still  water,  and  what  is  the  velocity  of  the  stream  ? 

45.  A  box  will  hold  18  quires  of  paper  and  18  bunches  of 
envelopes,  or  20  quires  of  paper  and  15  bunches  of  envelopes. 
How  many  quires  of  paper  will  the  box  hold?  How  many 
bunches  of  envelopes  will  it  hold? 


204  SIMULTANEOUS   SIMPLE  EQUATIONS 

46.  A  shelf  will  hold  20  arithmetics  and  24  algebras  or  15 
arithmetics  and  36  algebras.  How  many  arithmetics  will  the 
shelf  hold?     How  many  algebras  will  it  hold? 

47.  Two  men  had  a  certain  distance  to  row  and  took  turns  in 
rowing.  Whenever  the  first  rowed,  the  boat  moved  at  a  rate  suf- 
ficient to  cover  the  entire  distance  in  10  hours,  and  whenever  the 
second  rowed,  in  14  hours.  If  the  journey  was  completed  in  12 
hours,  how  many  hours  did  each  row  ? 

48.  A  train  ran  1  hour  and  36  minutes,  and  was  then  detained 
40  minutes.  It  then  proceeded  at  f  of  its  former  rate  and  reached 
its  destination  16  minutes  late.  If  the  detention  had  occurred  10 
miles  farther  on,  the  train  would  have  arrived  20  minutes  late. 
At  what  rate  did  the  train  set  out,  and  what  was  the  whole  dis- 
tance traveled  ? 

49.  A  certain  number  of  people  charter  an  excursion  boat, 
agreeing  to  share  the  expense  equally.  If  each  pays  a  cents, 
there  will  be  h  cents  lacking  from  the  necessary  amount ;  and  if 
each  pays  c  cents,  c^  cents  too  much  will  be  collected.  How  many 
persons  are  there,  and  how  much  should  each  pay  ? 

50.  A  sum  of  money  was  to  be  divided  equally  among  a  certain 
number  of  persons,  but  a  persons  more  than  were  expected  ap- 
peared to  claim  a  share,  and  in  consequence  each  received  b  dol- 
lars less.  If  there  had  been  c  persons  less  than  were  expected, 
each  would  have  received  d  dollars  more.  How  many  persons 
appeared,  and  how  much  did  each  receive  ? 

Give  the  results  when  a  =  ^,  b  =  100,  c  =  4,  and  d  =  125. 

51.  A  and  B  working  together  can  do  a  piece  of  work  in  a 
days.  But  finding  it  impossible  to  work  at  the  same  time,  A 
works  b  days,  and  later  B  finishes  the  work  in  c  days.  In  how 
many  days  can  each  do  the  work  alone  ? 

If  a  =  5j\,  b=:5,  and  c  =  6,  in  how  many  days  can  each  do 
the  work  alone  ? 

52.  A  purse  holds  c  crowns  and  d  guineas;   a  crowns  and  b 

guineas  will  fill  —  th  of  it.     How  many  will  it  hold  of  each  ? 
n 

How  many,  if  c  =  12,  d  =  6,  a  =  4,  6  =  6,  m  =  1,  and  w  =  2  ? 


SIMULTANEOUS   SIMPLE   EQUATIONS  205 

53.  A  mine  is  emptied  of  water  by  two  pumps  which  together 
discharge  m  gallons  per  hour.  Both  pumps  can  do  the  work  in  b 
hours,  or  the  larger  can  do  it  in  a  hours.  How  many  gallons  per 
hour  does  each  pump  discharge  ?  What  is  the  discharge  of  each 
per  hour  when  a  =  5,  6  =  4,  and  m  =  1250  ? 

54.  Two  trains  arc  scheduled  to  leave  A  and  B,  m  miles  apart, 
at  the  same  time,  and  to  meet  in  h  hours.  If  the  train  that  leaves 
B  is  a  hours  late  and  runs  at  its  customary  rate,  it  will  meet  the 
first  train  in  c  hours.     What  is  the  rate  of  each  train  ? 

What  is  the  rate  of  each,  if  m  =  800,  c  =  9,  a  =  If,  and  6  =  10  ? 

55.  If  a  quarts  of  good  wine  is  mixed  with  h  quarts  of  poorer 
wine,  the  mixture  will  be  worth  c  cents  a  quart;  if  h  quarts  of 
the  better  wine  is  mixed  with  a  quarts  of  the  poorer,  the  mixture 
will  be  worth  d  cents  a  quart.  What  is  each  kind  of  wine  worth 
per  quart  ?  What  is  each  kind  of  wine  worth  per  quart,  if  a  =  40, 
6  =  20,  c  =  100,  and  (^  =  80  ? 

212.  Discussion  of  the  general  solution  of  a  system  of  two  simul- 
taneous simple  equations  involving  two  unknown  numbers. 

Let  ax  +  hy  =  c  (1) 

and*  a'x-{-h'y  =  c'  (2) 

be  any  two  simultaneous  simple  equations. 

(1)  X  6',  ah'x  +  hh'y  =  h'c  (3) 

(2)  X  6,  a'hx  +  hh'y  =  he'  (4) 

(3)  _  (4),  {ab'-  a'h)x  =  h'c  -  he'  (5) 

(1)  X  a',  aa'x  +  a'hy  =  ca'  (6) 

(2)xa,  aa'x  +  ah'y  =  c'a  (7) 

(7)_(6),  {ah'-a'b)y  =  c'a-ca'  (8) 

By  the  principles  of  equivalence  the  given  system  may  be  replaced  by  the 
equivalent  system  (5,  8),  in  which  (5)  involves  x  alone  and  (8)  involves  y 
alone.  By  §  197,  eacli  of  these  simple  equations  involving  one  unknown 
number  has  one  and  only  one  root,  which  can  be  found  except  when  the 
common  coefficient  of  x  and  y  is  equal  to  zero.  Hence,  when  ah'  —  a'h  is  not 
equal  to  zero,  the  given  system  is  satisfied  by  one  and  only  one  set  of 

values  rl  r.    ,  ,  I 

^^Vc-hc^    and    y  =  ^-'«-««' 


ah'-  a'h  ab'  -  a'h 


*  In  algebraic  notation  a',  a",  a'",  etc.,  are  read  'a  prime,'  'a  second,' 
a  third,'  etc. 


206  SIMULTANEOUS   SIMPLE    EQUATIONS 

If  ah'  —  a'b  =  0,  that  is,  if  ab'=  a'b,  the  first  members  of  (3)  and  (4)  are 
identical,  and  therefore  the  second  members  must  be  equal.  The  same  is 
true  of  equations  (0)  and  (7).  Hence,  (4)  is  only  a  different  form  of  (3), 
and  (7)  is  only  a  different  form  of  (6);  that  is,  (1)  and  (2)  are  not  inde- 
pendent equations. 

But  if  ab'  —  a'b  is  not  equal  to  zero,  neither  (3)  and  (4)  nor  (6)  and  (7) 
are  reducible  to  the  same  form ;  that  is,  (1)  and  (2)  are  independent 
equations. 

It  is  evident  from  the  Distributive  Law  for  multiplication  that  the  equations 
(3)  and  (4),  and  also  (6)  and  (7),  cannot  be  combined  by  addition  or  sub- 
traction unless  X  and  y  have  the  same  values  in  (2)  as  in  (1) ;  that  is,  that 
(1)  and  (2)  cannot  be  solved  unless  they  are  simultaneous  equations. 

It  is  evident  that  equations  (5)  and  (8),  and  therefore  (1)  and  (2),  cannot 
be  solved  if  ab'—  a'b  =  0,  since  §  196,  Priu.  2,  the  members  cannot  be  divided 
by  a  known  expression  equal  to  0  ;  and  it  has  been  shown  that  (1)  and  (2) 
are  dependent  or  independent  equations  according  as  ab'  —  a'b  is  or  is  not 
equal  to  zero. 

Hence,  it  follows  that : 

Two  simple  equations  involving  two  unknown  numbers  cannot  be  solved 
unless  the  equations  are  simultaneous  and  independent. 

Every  system  of  two  independent  simultaneous  simple  equations  involving 
two  unknown  numbers  can  be  solved,  and  is  satisfied  by  one,  and  only  one, 
set  of  values  of  its  unknown  numbers. 


THREE   OR  MORE   UNKNOWN   NUMBERS 

213.  1.  In  the  equations  x  -\-2y  •{■  z  =  ^  and  2  x  -\-  y  —  z  =  l, 
how  may  z  be  eliminated  ? 

2.  If  one  of  the  unknown  numbers  in  the  above  equations  is 
eliminated,  how  many  unknown  numbers  will  be  left  ? 

3.  How  many  independent  equations  are  necessary  before  the 
values  of  ^wo  unknown  numbers  can  be  found  ? 

4.  How  many  independent  equations  containing  the  same  two 
unknown  numbers  can  be  formed  by  combining  the  equations 
m(l)? 

5.  Since  only  one  derived  equation  containing  two  unknown 
numbers  was  obtained  from  the  given  equations  by  eliminating  z, 
and  since  we  must  have  two  such  equations  before  we  can  find 
the  values  of  x  and  ?/,  if  another  independent  equation  involving 
Xj  y,  and  z  is  combined  with  either  of  the  equations  in  (1),  how 


SIMULTANEOUS   SIMPLE  EQUATIONS  207 

many  independent  equations  containing  x  and  y  only  will   be 
available  for  finding  the  values  of  x  and  y? 

6.  When  the  values  of  x  and  y  are  found,  how  may  the  value 
of  z  be  found  ? 

7.  Then,  how  many  independent  equations  containing  three 
unknown  numbers  must  be  given,  so  that  the  values  of  the  un- 
known numbers  may  be  found  ?  How  many  to  find  the  values 
of  four  unknown  numbers  ? 

214.  Principle.  —  Every  system  of  independent  simultaneous 
simple  eqvMions  involving  the  same  number  of  unknown  numbers  as 
there  are  equations  can  be  solved,  and  is  satisfied  by  one  and  only 
one  set  of  values  of  its  unknown  numbers. 

The  above  principle  may  be  established  as  follows : 

From  the  given  system  of  n  equations  involving  n  unknown  numbers,  a 
second  system  of  n  —  1  equations  involving  n  —  \  unknown  numbers  may  be 
derived  by  eliminating  one  of  tlie  unknown  numbers  ;  from  the  second  system 
a  third  system  of  w  —  2  equations  involving  n  —  2  unknown  numbers  may  be 
derived  ;  and  this  process  may  be  continued  until  the  nth  system,  a  single 
simple  equation  involving  only  one  unknown  number,  is  obtained. 

Since,  §  197,  this  equation  has  one  and  only  one  root,  by  substituting  this 
value  in  either  of  the  two  equations  of  the  next  preceding  system  and  solving, 
one  and  only  one  value  of  the  other  number  in  that  equation  is  obtained  ;  by 
substituting  these  two  values  in  any  one  of  the  three  equations  of  the  next 
preceding  system,  one  and  only  one  value  of  the  remaining  unknown  number 
in  that  equation  is  obtained  ;  and  by  continuing  this  process,  the  value  of 
each  of  the  other  unknown  numbers  is  obtained. 

By  the  principles  of  equivalent  equations,  the  following  system  of  n  equa- 
tions may  be  substituted  for  the  given  system  :  the  single  equation  finally 
derived  by  elimination  and  composing  the  nth  system  ;  either  of  the  two 
equations  of  the  preceding,  or  (n  —  l)th  system  ;  any  one  of  the  three  equa- 
tions of  the  system  preceding  that,  or  of  the  («  —  2)th  system  ;  and  so  on  to 
any  one  of  the  n  equations  in  the  1st  or  given  system. 

But  each  of  the  n  equations  just  described  has  one  and  only  one  value 
of  an  unknown  number.  Hence,  the  given  system  can  be  solved,  and  is 
satisfied  by  one  and  only  one  set  of  values  of  its  unknown  numbers. 

If  the  number  of  unknown  numbers  is  greater  than  the  number  of  inde- 
pendent simultaneous  equations^  the  last  equation  obtained  by  repeated  elimi- 
nations is  indeterminate,  and  hence  the  system  is  indeterminate. 

If  the  number  of  unknown  numbers  is  less  than  the  number  of  independent 
simultaneous  equations,  say  n  —  p,  any  n  —  p  of  the  equations  involving  the 
n  —  p  unknoivn  numbers  form  a  determinate  system. 


208  ACADEMIC   ALGEBRA 

Examples 

1.    Find  the  values  of  x,  y^  and  z  in      2  a;  -f-  2/  +  2  2  ==  10, 

307  +  4?/ -32  =  2. 

Solution 
X +  2^  +  82;  =  14  (1) 

2x  +  y  +  2z  =  10  (2) 

Sx  +  iy-Sz  =  2  (3) 

Eliminating  z  by  combining  (1)  and  (3), 

(1)  +  (3),  4a; +  6^=16  (4) 
Eliminating  z  by  combining  (2)  and  (3), 

(2)  X  3,  6  a;  +  3  ?/  +  6  0  =  30  (5) 
(3)x2,                                     6x  +  Sy  -6z=    4:                                          (6) 

Adding,                                       12  x  +  11  y  =  34  (7) 
Eliminating  x  by  combining  (7)  and  (4), 

(4)x3,                                          12«4-18?/  =  48  (8) 

(8) -(7),                                                       1y  =  U  (9) 

/.  2/  =  2  (10) 

Substituting  in  (4),                       4  sc  +  12  =  16  (11) 

.-.  X  =  1  (12) 
Substituting  the  values  of  x  and  ?/  in  (1), 

1  +  4  +  3  ^  =  14  (13) 

.-.  z  =  3  (14) 

Explanation.  —  Eliminating  z  from  (1)  and  (3)  and  from  (2)  and  (3), 
two  simultaneous  equations,  (4)  and  (7),  are  obtained  involving  x  and  y. 
By  the  principle  of  elimination,  §  208,  the  new  system  (1,  4,  7),  or  (2,  4,  7), 
or  (3,  4,  7),  is  equivalent  to  the  given  system. 

Eliminating  x  from  (4)  and  (7),  a  simple  equation  involving  but  one 
unknown  number  y  is  obtained,  and  from  this  equation  the  value  of  y  is 
found,  equation  (10).  Hence,  the  system  (1,  4,  7)  has  been  replaced  by  the 
equivalent  system  (1,  4,  10),  which  is,  therefore,  equivalent  to  the  given 
system. 

Substituting  2  for  y  in  (4),  the  value  of  x  is  found,  giving  a  new  system 
(1,  12,  10)  equivalent  to  the  given  system.  Substituting  the  values  of  both 
X  and  y  in  (1),  the  value  of  z  is  found,  giving  the  desired  system  (14,  12,  10) 
equivalent  to  the  given  system. 


SIMULTANEOUS   SIMPLE   EQUATIONS 


209 


4. 


5. 


7. 


8. 


10. 


Solve  the  following: 


2x-  y-\-2z  =  12j 
x-^3y+  2  =  41, 

2x^y  +  Az  =  22. 

Sx-\-5y  —  z  =  S, 
4a; +  3?/ +  22  =  47, 
6x-{-5y-2z  =  ll. 

x-tSy—z  —  10, 
2  a; +  5?/ +  42  =  57, 
Sx-y  +  2z  =  15. 

a;  +  ?/  +  2  =  53, 

a; +  22/ +  32  =  105, 

a;  +  3y  +  42  =  134. 

X  —  y  -\-  z  =  30, 
Sy-x-z  =  12, 
72  —  2/  +  2a;=  141. 

8a;-52/  +  2z  =  53, 
a;  +  2/  -  2  =  9, 
13a;- 9?/ +  32  =  71. 

a;  +  3?/  +  42  =  83, 

a;  +  2/  +  2  =  29, 

6a;  +  82/  +  32  =  156. 

2a; +  32/ +  42  =  29, 
3a; +  22/ +52  =  32, 

4a; +  32/ +  22  =  25. 

2 X  —  3 y  -{-  4:Z  —  V  =  4:j 
4:X-[-2y  —  z  +  2v  =  13, 
a;  —  2/  +  22  +  3v  =  17, 
3a;  +  22/-z  +  4v  =  20. 


11. 


12. 


13. 


14. 


15. 


16. 


17. 


18. 


l3x-2y  +  z  =  2, 

2a;  +  52/  +  22  =  27, 
a;  +  32/  +  32  =  25. 

'4a;  —  52/  +  32  =  14, 

a; +  72/ -2  =  13, 
.2a;  +  52/  +  52  =  36. 

2x  +  y  —  3z-\-4:io  =  M, 
3x  —  2y-\-z  —  w  =  —  lf 
4a;  —  2/  +  22  +  «;  =  55, 
5a;-3y  +  42-w  =  39 

7a;-l  =  32/, 
11  2  —  1  =  7  V, 
42-1  =  72/, 
19a;-l=3v. 

a?  +  i2/  +  i^  =  32, 
ia;  +  i2/  +  i2  =  15, 
i^  +  i2/  +  i^  =  12. 

ria;-i2/  +  i2;  =  3, 

ix-\y-{-lz  =  l, 

Iia;-i2/  +  i2;  =  5. 

^  +  32  =  29, 

^^  +  22  =  22, 
3a; -2/ =  3(2-1). 

[3x-\-y  —  z-\-2v  =  0, 
3y-2x  +  z-4:V  =  21, 
x  —  y  +  2z  —  3v  =  6, 
4.x  +  2y-3z-\-v  =  12. 


ACAD,  ALG. 


14 


210 


ACADEMIC  ALGEBRA 


u-\-v-\-x—y-\-z=l, 
19.   Solve  \  u-\-v—x^y-^z=^, 

u—v-\-x-\-y-\-z=llf 
v—u-\-x-\-y-\-z=l^. 
Solution.  —  Adding  the  equations, 

Dividing  by  3,  u  +  v-[-x  +  y  +  z  =  \b. 

Subtracting  each  of  the  given  equations  from  this  equation, 
20  =  10,    2y  =  8,    2x  =  6,    2v  =  4,    2m  =  2; 


.-.  5=5,         y 

Solve  the  following : 

f  a;  +  2/  =  9, 
y+z  =  l, 
z  -{■  x  =  Z. 
V  -\-x  -\-y=  15, 
x-\-y  +  z  =  l^, 
y  +  z-\-v  =  ll, 

Z  -\- V  -\- X  —  IQ. 


X      y 


x  =  3. 


20 


21. 


24. 


25. 


22.    ^i  +  i^ 

y     2 

Z        X 


xy 


23. 


x  +  y 

yz 
y  +  z 

zx 


z  +  x 
Suggestion.  —  If 


10, 

:8. 
1 

'5' 
1 

1 

''l' 


xy    _l 

at  +  2/  ~  5' 


xy 


=  2,       w  =  1. 
x+Sy+z 

X-^y  +  Sz 

[Sx  +  y-hz 
y-^z  +  v  — 
z-\-v  -\-  x  — 
v+x+y— 
x-\-y-\-z- 

1+1-1= 

X      y 

1  +  ^3  = 

y    ^ 

1+1-2= 

Z       X 

a; +  2/  a 
yz    _1 

2/ +  2  &' 
ga;    _1 

0  -4-  a;      c 


y      5        ^  lie 

—  —  -  ;   whence,  -  +  -  =  6. 

1'  y    X 


26. 


=  14, 
=  16, 
=  20. 

a;  =  22, 
2/ =  18, 
■  2  =  14, 
v  =  10. 

0, 
0, 


27 


SIMULTANEOUS  SIMPLE  EQUATIONS 


211 


- 

X 

b 

y 

+r«'     . 

28.    Solve 

hzx 

— 

cxy  H-  ayz  =  bxyz, 

a 

b 

-5  =  c. 

X 

y 

z 

Solution 

(l)  +  (3), 

2«  =  a  +  c. 

X 

a  +  c 

(2)^xyz, 

y     z     X 

(5) -(3), 

—  =  b  —  c. 

y 
.'.y=  '^. 

Substituting  the  values  of  x  and  y  in  (1),  and  solving, 

.=    2«   . 

(1) 

(2) 
(3) 


(4) 
(6) 


29. 


30. 


31. 


32. 


axy  —  x  —  y  =  0, 

bzx  —  z  —  X  =  Of 

.  cyz  —  y  —  z  =  0. 

i  x-\-y-z  =  0, 
\x  —  y  =  2b, 
[x-{-z  =  3a-\-b. 

V  -\-x  =  2  a, 
x-\-y  =  2a-z, 
y  +  z  =  a-\-b, 
[  V  —  z  =  a-^c. 

y-\-z  —  3x  =  2a, 

z-\-x-3y  =  2b, 

x-\-y  —  3z  =  2Gy 

.2x  +  2y  +  v  =  0. 


a  —  b 


33. 


34. 


35. 


36. 


abxyz-\-ca^—ayz—hzx=0, 
bcxyz  -\-  ayz — bzx — cxy = 0, 
caxyz  -h  bzx — cxy — ayz = 0. 

x-^y  +  z  =  a-\-b  +  c, 
x-\-2y  +  3z  =  b-\-2c, 
x-\-3y-i-4:Z  =  b-{-3c. 

v-^x-{-y  =  a-\-2b  +  c, 
x-{-y-[-z  =  3b, 
y  -\-z  -\-v  =  a-\-b, 
z-\-v-{-x  =  a-\-3b  —  c. 

ax -{- by  -\- cz  =  3, 
a-\-b 


a;  4-2/  = 
y  +  z  = 


ab 

b-\-c 
be 


212  '  ACADEMIC  ALGEBRA 

» 
Problems 

^  215.  1.  Three  men  bought  grain  at  the  same  prices.  A  paid 
$  4.80  for  2  bushels  of  rye,  3  bushels  of  wheat,  and  4  bushels  of 
oats ;  B  paid  $  6.40  for  3  bushels  of  rye,  5  bushels  of  wheat,  and 

2  bushels  of  oats;  and  C  paid  $5.30  for  2  bushels  of  rye,  4 
bushels  of  wheat,  and  3  bushels  of  oats.  What  was  the  price  of 
each? 

2.  A  dealer  was  asked  his  price  for  10  bushels  of  wheat,  corn, 
and  rye.  He  replied,  "  For  5  of  wheat,  2  of  corn,  and  3  of  rye, 
$  6.60 ;  for  2  of  wheat,  3  of  corn,  and  5  of  rye,  $  5.80 ;  and  for 

3  of  wheat,  5  of  corn,  and  2  of  rye,  $  5.60."  What  prices  had 
he  in  mind  ? 

3.  Divide  90  into  three  parts  such  that  the  sum  of  i  of  the 
first,  ^  of  the  second,  and  J  of  the  third  shall  be  30 ;  and  the 
first  shall  be  twice  the  third  diminished  by  twice  the  second. 

4.  There  are  three  numbers  such  that  the  sum  of  I-  of  the 
first,  i  of  the  second,  and  i  of  the  third  is  12 ;  of  ^  of  the  first, 
I  of  the  second,  and  ^  of  the  third  is  9,  and  the  sum  of  the  num- 
bers is  38.     What  are  the  numbers  ? 

5.  There  are  three  numbers  whose  sum  is  72.  If  the  sum  of 
the  first  two  is  divided  by  the  third,  the  quotient  is  If ;  and  if 
the  third  is  subtracted  from  twice  the  first,  the  remainder  will  be 
^  of  the  second.     Eind  the  numbers. 

y  6.  A  and  B  can  do  a  piece  of  work  in  10  days ;  A  and  C  can 
do  it  in  8  days ;  and  B  and  C  can  do  it  in  12  days.  How  long 
will  it  take  each  to  do  it  alone  ? 

7.  A  certain  number  is  expressed  by  three  digits  whose  sum  is 
14.  If  693  is  added  to  the  number,  the  digits  will  appear  in 
reverse  order.  If  the  units'  digit  is  equal  to  the  tens'  digit 
increased  by  6,  what  is  the  number  ? 

^  8.  The  third  digit  of  a  number  of  three  digits  is  as  much  larger 
than  the  second  digit  as  the  second  is  larger  than  the  first.  If 
the  number  is  divided  by  the  sum  of  its  digits,  the  quotient  is  15. 
AVhat  is  the  number,  if  the  order  of  its  digits  may  be  reversed  by 
adding  396  ? 


SIMULTANEOUS  SIMPLE  EQUATIONS  213 

9.  Find  three  numbers  such  that  the  first  increased  by  ^  of 
the  sum  of  the  other  two  shall  be  36 ;  the  second  increased  by  \ 
of  the  sum  of  the  other  two  shall  be  40 ;  and  the  third  increased 
by  \  of  the  sum  of  the  other  two  shall  be  44. 

10.  Divide  800  into  three  parts  such  that  the  sum  of  the  first, 
I  of  the  second,  and  -|  of  the  third  shall  be  400 ;  and  the  sum  of 
the  second,  f  of  the  first,  and  \  of  the  third  shall  be  400. 
,  11.  Three  cities.  A,  B,  and  C,  connected  by  straight  roads,  are 
situated  at  the  vertices  of  a  triangle.  From  A  to  B  by  way  of  C 
is  130  miles ;  from  B  to  C  by  way  of  A  is  110  miles ;  and  from  C 
to  A  by  way  of  B  is  140  miles.     How  far  apart  are  the  cities  ? 

12.  Find  three  numbers  such  that  the  first  with  \  of  the  sum  of 
the  second  and  third  is  340  ;  the  second  with  ^  of  the  sum  of  the 
first  and  third  is  600;  and  the  third  with  i  of  the  remainder 
when  the  first  is  subtracted  from  the  second  is  450. 

13.  A  merchant  has  three  kinds  of  tea.  He  can  sell  2  pounds  of 
the  first  kind,  3  of  the  second,  and  4  of  the  third  for  $  4.70 ;  or  4 
of  the  first,  3  of  the  second,  and  2  of  the  third  for  $  4.30.  If  a 
pound  of  the  third  kind  is  worth  5  cents  more  than  f  of  a  pound 
of  the  first  kind  and  ^  of  a  pound  of  the  second  kind,  what  is  the 
value  of  1  pound  of  each  kind  ? 

14.  A,  B,  and  C  have  certain  sums  of  money.  If  A  gives  B 
%  100,  they  will  have  the  same  amount ;  if  A  gives  C  ^  100,  C 
will  have  twice  as  much  as  A;  and  if  B  gives  C  ^100,  C  will 
have  4  times  as  much  as  B.     What  sum  has  each  ? 

15.  A  quantity  of  water  sufficient  to  fill  three  jars  of  different 
sizes  will  fill  the  smallest  jar  4  times ;  the  largest  jar  twice  with 
4  gallons  to  spare ;  or  the  second  jar  3  times  with  2  gallons  to 
spare.    What  is  the  capacity  of  each  jar  ? 

^  16.  A  gave  to  B  and  C  as  much  as  each  of  them  had ;  B  then 
gave  to  A  and  C  as  much  as  each  of  them  had ;  and  C  then  gave 
to  A  and  B  as  much  as  each  of  them  had,  after  which  each  had 
$  8.     How  much  had  each  at  first  ? 

17.  Three  boys.  A,  B,  and  C,  each  had  a  bag  of  nuts.  After 
each  boy  had  given  each  of  the  others  \  of  the  nuts  in  his  bag, 
they  counted  and  found  that  A  had  740,  B  580,  and  C  380.  How 
many  had  each  at  first  ? 


INVOLUTION 


216.  1.  How  many  times  is  a  number  used  as  a  factor  in  pro- 
ducing its  second  power  ?  its  third  power  ?  its  fourth  power  ?  its 
fifth  power  ?  its  nth  power,  when  w  is  a  positive  integer  ? 

2.  What  is  the  meaning  of  2^?  of  (-2/?  of  a'?  of  (axy? 
of  of,  when  n  is  a  positive  integer  ? 

3.  What  sign  has  (+  a)^  ?  (+  a)^  ?  (+  ay,  or  any  power  of  a  ? 
What  sign  has  any  power  of  a  positive  number  ? 

4.  What  sign  has  (-a)2?  (-a)»?  (-a)^?  {-a)'? 

What  sign  have  the  even  powers  of  a  negative  number  ?  What 
sign  have  the  odd  powers  ? 

5.  What  is  the  fourth  power  of  a^  ?  of  a^  ?  of  a^«  ?  of  a",  when 
w  is  a  positive  integer  ?  What  are  the  fifth  powers  of  these  num- 
bers ?  the  sixth  powers  ?  the  mth  powers,  when  m  is  a  positive 
integer  ? 

6.  How  does  8^  compare  in  value  with  2^  x  4^  ?  with  2^  x  2^  x  2^  ? 
32  with  62  --  22  ?  52  with  10^  --  2'  ? 

217.  The  process  of  finding  any  required  power  of  an  ex- 
pression is  called  Involution. 

218.  Principles.  — 1.  Law  of  Signs. — All  powers  of  a  positive 
number  are  positive;  even  powers  of  a  negative  number  are  positive, 
and  odd  powers  are  negative. 

2.  Law  of  Exponents. — TTie  exponent  of  a  power  of  a  number  is 
equal  to  the  exponent  of  the  number  multiplied  by  the  exponent  of  the 
'power  to  which  the  number  is  to  be  raised. 

3.  Any  power  of  a  product  is  equal  to  the  product  of  its' factors 
each  raised  to  that  power. 

4.  Any  power  of  the  quotient  of  two  numbers  is  equal  to  the  quo- 
tient of  the  numbers  each  raised  to  that  power. 

214 


INVOLUTION  215 

The  above  principles  may  be  established  as  follows : 
Principle  1  follows  directly  from  the  law  of  signs  for  multiplication. 
Principle  2.     When  m  and  n  are  positive  integers, 
§  24,  (a")"  =  a"*  X  a*"  X  a"*  •••  to  n  factors 

— -  Qm+m-\-m+—  to  n  terms 

=  a*"". 
Principle  3.     When  w  is  a  positive  integer, 
§  24,  (a&)'»=  ab  X  ab  X  ab  •••  to  w  factors 

§  83,  ={aaa  •••  to  n  factors)  {bbb  •••  to  n  factors) 

=  a"6". 
Principle  4.     When  w  is  a  positive  integer, 

§24,  f?V  =  ?x-x?...  towfactore 

\o I        b      h      b 

aaa  •••  to  w  factors 


180, 


bbb  "-ton  factors 
6«* 


219.   Involution  of  monomials. 

Examples 

1.  What  is  the  third  power  of  4  o?h  ? 
Solution.         (4  a^b)^  =  4  a^6  x  4  a^b  x  4  d^b  =  64  a^b^. 

2.  What  is  the  fifth  power  of  -  2  ab^  ? 

Solution.     (  -  2  ab'^y  z=-2ab^  x -2ab'^  x -2ab'^  x -2  ab'^  x  -  2  a&2 
=  -  32  a^&io. 

To  raise  an  integral  term  to  any  power : 

EuLE.  —  liaise  the  numerical  coefficient  to  tJie  required  power  and 
annex  to  it  each  letter  ivith  an  exponent  equal  to  the  product  of  its 
exponent  by  the  exponent  of  the  required  power. 

Prefix  the  sign  +  to  any  poiver  of  a  positive  number  or  to  an  even 
power  of  a  negative  number;  the  sign  —  to  an  odd  power  of  a  nega- 
tive number. 


216 


ACADEMIC  ALGEBRA 


To  raise  a  fraction  to  any  power : 

Rule.  —  liaise  both,  numerator  and  denominator  to  the  required 
power  and  prefix  the  proper  sign  to  the  result. 

Eaise  to  the  power  indicated : 


3. 

(a6V)2. 

15.    ( 

'abexy. 

4. 

(a?hh)\ 

16.    ( 

2  e'aff. 

5. 

(2a^cy. 

17.    ( 

[3  bey. 

6. 

(laVf, 

18.    ( 

;2aV/. 

7. 

(-ly. 

19.    ( 

;-ir- 

8. 

X-ah)\ 

20.    ( 

-1)"^. 

9. 

i-Scf. 

21.    ( 

;-i)- 

10. 

(-lOoj^f. 

22.    ( 

^-  by+\ 

11. 

(-Qa'icy. 

23.    ( 

;_  62)2n+l^ 

12. 

(-4cy)^ 

24.    ( 

;-  a^byc'-Hy. 

13. 

{-2lVdy. 

25.    ( 

'-a^-ifpz^y. 

14. 

(-  aV2/"-^)2. 

26.    ( 

;-a"-i6"-^c)3. 

27. 

What  is  the  square  of  — 

5aV^ 

Tft^c* 

Solution 

[      7  h-^c  I 

5_     hopx'^         hd^y? 
7  hH           7  &2c 

25«6«* 
49  64c2 

Raise  to  the  power  indicated  • 

• 

28. 

& 

33.    (- 

»•  (-¥)• 

29. 

©•• 

"■(- 

.AY. 

"■  (-f)' 

30. 

[lOfJ 

35.    {- 

3a;Y 

•'•  (S)-- 

31. 

f2xy 

36.    {- 

2aY 

"■  (^;)-- 

32. 

(  ^^  Y- 

V2  6«-V 

37.    (- 

-•  (SSs)- 

INVOLUTION  217 

220.  Involution  of  polynomials. 

§  91,  (a  +  6)2  =  a2  +  2  a6  +  &2. 

§  93,  (a  -  6)2  =  a2  -  2  a6  +  62. 

§  95,  (a  +  6  +  c)2  =  a2  _^  62  +  c2  4.  2  a6  +  2  ac  +  2  6c. 

Raise  the  following  to  the  second  power : 

1.  2a-f6.                    5.    3a;  — 4?/^.  9.  a  — 6  +  ^  — y. 

2.  2a— 6.                     6.    Sm'*- 11.  10.  a"*  +  a.-'' —  2/"^^ 

3.  a"*- 3  6".                 7.    1  — 3a6c.  11.  2a +  36  — 4c. 

4.  a2-2a:2„.                 8.    4a;^  +  5.  12.  5a2-lH-4n3. 

Raise  to  the  required  power  by  multiplication : 

13.  (x  +  yf.  15.    {x-\-y)\  17.    (.r-f-i/)*. 

14.  (x  —  yf.  16.    («  —  ?/)*.  18.    (x  —  yy. 

221.  Involution  of  binomials  by  the  Binomial  Theorem. 
By  multiplication, 

(a  +  xY=a^  +  Sa^x-\-Sa:x?-{-a?. 
{a-xf=a?-Sa^x  +  Zax'-7?. 
(a  +  a;)*  =  a*  +  4  a^a;  +  6  a^ar^  +  4  aa;^  +  ar*. 
(a  —  xy  =  a*  —  4:  a^x  +  6  a^x^  —  4  oa:^  +  x*. 
(a  +  a;y=  a*  4-  5  a^a;  +  10  a-V  +  10  aV  +  5  aa;*  +  x^. 
(a  -  xf  =  a^  -  5  a^a;  +  10  aV  -  10  a V  +  5  aa;*  -  a^. 
Examine  carefully  the  above  powers  of  (a  +  x)  and  (a  —  x). 

1.  How  does  the  number  of  terms  in  a  power  of  a  binomial 
compare  with  the  exponent  of  the  binomial  ? 

2.  What  terms  of  the  power  contain  the  first  term  of  the  bino- 
mial ?  the  second  term  of  the  binomial  ?  both  terms  ? 

3.  What  is  the  exponent  of  the  first  term  of  the  binomial  in 
the  first  term  of  the  power  ?  in  the  second  ?  in  the  third,  etc.  ? 


218  ACADEMIC  ALGEBRA 

4.  What  is  the  exponent  of  the  second  term  of  the  binomial  in 
the  second  term  of  the  power  ?  in  the  third  ?  in  the  fourth,  etc.  ? 

5.  What  is  the  coefficient  of  the  first  term  of  the  power? 
How  does  the  coefficient  of  the  second  term  compare  with  the 
exponent  of  the  binomial  ? 

6.  If  the  coefficient  of  any  term  is  multiplied  by  the  exponent 
of  the  first  term  of  the  binomial  found  in  that  term,  and  the 
product  is  divided  by  the  number  of  the  term,  how  does  the 
quotient  compare  with  the  coefficient  of  the  succeeding  term  ? 

7.  What  are  the  signs  of  the  terms  in  any  power  of  (a-\-  b)? 
What  terms  are  negative  in  any  power  of  (a—b)? 

222.  Principles.  —  1.  The  number  of  terms  in  a  positive  in- 
tegral  power  of  a  binomial  is  one  greater  thayi  the  index  of  the 
required  power. 

2.  Tlie  first  term  of  the  power  contains  only  the  first  term  of  the 
binomial;  the  last  term  of  the  power,  only  the  second  term  of  the 
binomial;  all  other  terms  of  the  power  contain  as  factors  both  terms 
of  the  binomial. 

3.  The  exponent  of  the  first  term  of  the  binomial  in  the  first  term 
of  the  power  is  the  same  as  the  index  of  the  required  poiver,  and  it 
decreases  1  in  each  succeeding  term.  The  exponent  of  the  second 
term  of  the  binomial  in  the  second  term  of  the  power  is  1,  ajid  it 
increases  1  in  each  succeeding  term. 

4.  T7ie  coefficient  of  the  first  term  of  the  power  is  1.  The  co- 
efficient of  the  second  term  is  the  same  as  the  index  of  the  required 
power. 

5.  The  coefficient  of  any  term  may  be  found  by  multiplying  the 
coefficient  of  the  preceding  term  by  the  exponent  of  the  first  term  of 
the  binomial  found  in  that  term,  and  then  dividing  the  result  by  the 
number  of  the  term. 

6.  If  both  terms  of  the  binomial  are  positive,  all  the  terms  of  any 
power  of  the  binomial  will  be  positive. 

7.  If  the  second  term  of  the  binomial  is  negative  and  the  first  term 
positive,  the  terms  of  any  power  of  the  binomial  will  be  alternately 
positive  and  negative. 


INVOLUTION 


219 


Examples 
1.   Find  the  fifth  power  of  (6  —  y)  by  the  binomial  theorem. 

Solution 

Letters  and  exponents,        h^        ¥y  b^y^  h^y^        by^      y^ 

Coefficients,  15  10  10  5  1 

Signs,  _|.         _  +  _  +  _ 


•Combined, 


65  _  5  ffy  ^  10  i)Sy2  _  10  b-2y^  +  5  62/4  -  y^ 


In  every  term  of  a  power  of  a  binomial  the  sum  of  the  exponents  of  the 
terms  of  the  binomial  is  equal  to  the  index  of  the  required  power. 

Expand : 


2. 

(x  +  yy. 

13. 

(c-ny. 

24. 

(x-2y. 

3. 

(m  +  ny. 

14. 

(x-ay. 

25. 

(x-^ry. 

4. 

[m  —  ny. 

15. 

(d-yy. 

26. 

(b-cy. 

5. 

{a-cf. 

16. 

{b  +  yy- 

27. 

(p-^-qy^ 

6. 

{a  +  by. 

17. 

(m  +  ny. 

28. 

(a -by. 

7. 

{h  +  ay. 

18. 

(p-gy- 

29. 

(a  +  bey. 

8. 

(q-ry. 

19. 

(s-hty. 

30. 

{ab  -  cy. 

9. 

{c  +  dy. 

20. 

(x  +  2y. 

31. 

(m-pny. 

10. 

(x-hyf. 

21. 

(a +  3)1 

32. 

(m  —  any. 

11. 

(x-yy. 

22. 

(x  +  4.y. 

33. 

(ax  —  byy. 

12. 

{x-yy. 

23. 

(x-\-5y. 

34. 

(ax  -  byy. 

35.    Expand  {2b^-Zyy. 


Solution 
Let  2  62  =  fl5^  and  3  y  =  a;. 

Then,  2  62  -  .3y  =  a  -  «, 

and  (2  62-3?/)*  =  (a-x)* 

=  a*  -  4  a^a;  +  6  a'^x'^  -  4  ax^  +  «* 
Restoring  values,  =  (2  62)*  -  4  (2  62)3(3  y)  +  6  (2  62)2(3  y)^ 

-4(2  62)(3?/)3+(32/)* 
=  16  68  -  96  6«y  +  216  6*y2  _  216  b-y^  +  81  y* 


220  ACADEMIC  ALGEBRA 

36.    Expand  (1  +  a^)^- 


Solution 

(1  +  x^r 

=  13 +  3(1)2(^:2)+ 3(1)  (a: 

;2)2+(a 

:2)8 

Expand : 

37.    (x  +  2yy. 

41. 

(1-3^)^ 

45. 

(1-xy. 

38.    (2x-y)\ 

42. 

{px^-dbf. 

46. 

{l-2xf. 

39.    (2x-5f. 

43. 

(1  +  a^fe^)^ 

47. 

(^-i/. 

40.    (o^-lOy. 

44. 

(2aa;-&y. 

48. 

a^-i2//. 

Expand : 

«■■  (^•+i)'- 

52. 

(--I)- 

55. 

(f.--j 

-  (r9- 

53. 

(-¥)• 

56. 

e-)" 

"•  e-9' 

54. 

(i-¥)' 

57. 

(-=)■• 

58.  Expand  {a-h-  cf. 

Solution 

(a  — 6  — c)3=(a  — 6-c)3,  a  binomial  form. 

(«^-c)3=(a-6)3_3(a-&)2c+3(a-6)c2-c3 

=a3-3a25+3a&2_ft3_3c(a2_2rt6  +  ?)2)  +  3ac2-3  6c2-c3 
=a3-3 a25  +  3  ah'^-h^-^  a2c+6  a&c-3  62c+3  ac2-3  6c2-c8. 

59.  Expand  (a  +  &  —  c  —  cr)^. 


Suggestion.      {a  +  h  -  c  -  dy  ={a  +  h  -  c  -{■  dy,  a  binomial  form, 

Expand : 

60.  (a  +  x-y)\  66.  (a4-26-3c)^ 

61.  {a-m-n)\  67.  (a  +  6  +  a;  +  2/)^. 

62.  {a—x  +  yf.  68.  (a  +  6  — a;  — y)^ 

63.  (a-x-yf.  "    69.  (a  -  &  +  a; -  y)^. 

64.  (a  +  a;  +  2)3.  70.  (a-b-x-^yf. 

65.  (a-a;-2)».  71.  {a-h-x-yf. 


EVOLUTION 


223.  1.  Of  what  two  equal  numbers  is  16  the  product  ?  What 
is  the  square  root  of  16  ?  Since  16  is  equal  also  to  —  4  x  —  4, 
what  other  square  root  may  16  have  ?  What  is  the  square  root 
of  25  ?  of  64  ?     What  is  the  fourth  root  of  16  ?  of  81  ? 

2.  What  is  the  sign  of  an  even  root  of  a. positive  number? 

4 

3.  Can  the  square  root  of  -  16  be  found  ?  of  -  25  ?  of  -  64? 
the  fourth  root  of  —  16  ?  of  —  81  ?  Can  an  even  root  of  any 
negative  number  be  found  ? 

4.  What  is  the  cube  root  of  8?  of  27  ?  of  64?  of  -8? 
of  -  27  ?  of  -  64  ?     What  is  the  fifth  root  of  32  ?  of  -  32  ? 

5.  How  does  the  sign  of  an  odd  root  of  a  number  compare 
with  the  sign  of  the  number  ? 

6.  Since  a^=o?  x  a^  X  a^,  what  power  of  a^  is  a^?  What  is 
the  cube  root  of  a^  ?  of  a^  ?  of  a^  ?  How  is  the  exponent  of  a  in 
the  cube,  or  third,  root  of  any  power  of  a  found  ?  What  is  the 
fourth  root  of  a8?  of  a^? 

7.  How  is  the  exponent  of  a  root  of  a  power  obtained  from 
the  index  of  the  power  and  the  index  of  the  root  ? 

8.  How  does  V4  x  25  compare  in  value  with  VixV^? 
Vr>r9  with  Vi  X  V9 ?  \/8  X  1000  with  ^8  x  -5^1000?  In  each 
case  how  does  the  root  of  the  product  compare  in  value  with  the 
product  of  the  roots  of  the  factors  ? 

9.  How  does  Vl00-^4  compare  in  value  with  A/lOO-^V4? 
V36"T9  with  V36  -  V9  ?  -V/64T8  with  -^64  - ^  ?  In  each 
case  how  does  the  root  of  the  quotient  compare  in  value  with  the 
quotient  of  the  roots  of  the  dividend  and  the  divisor  ? 

221 


222  ACADEMIC  ALGEBRA    . 

224.  The  process  of  finding  any  required  root  of  an  expression 
is  called  Evolution. 

225.  Since  the  product  of  two  numbers  having  like  signs  is 
positive,  every  positive  number  has  two  square  roots,  numerically 
equal,  but  with  opposite  signs.  It  will  be  seen  later  that  every 
number  has  two  square  roots,  three  cube  roots,  four  fourth  roots, 
five  fifth  roots,  and,  in  general,  q  qih  roots.  Of  these  roots  the 
positive  roots  of  positive  numbers  and  the  negative  odd  roots  of 
negative  numbers  are  called  Principal  Roots. 

V25  =  +  5  or  —  5,  and  +  5  is  the  principal  square  root.  VlQ  =  +  2  or 
—  2  or,  as  will  be  seen  later,  +  V—  4  or  —  V—  4,  but  +  2  is  the  principal 
root.     The  principal  cube  root  of  8  is  +  2  and  of  —  8  is  —  2. 

226.  A  number  that  is  or  can  be  expressed  as  an  integer  or  as 
a  fraction  with  integral  terms  is  called  a  Rational  Number. 

a,  3,  5|^,  a^  +  b^,  \/25,  and  .338  are  rational  numbers. 

A  number  that  cannot  be  expressed  as  an  integer  or  as  a  frac- 
tion with  integral  terms  is  called  an  Irrational  Number. 

When  the  indicated  root  of  a  number  cannot  be  exactly  obtained, 
the  root  is  irrational. 

The  indicated  roots  \/2,  \/4,  Vcfi~+~F^,  v^,  v^,  and,  in  general,  the 
qth  root  of  a  number  that  is  not  the  qth  power  of  some  rational  number,  are 
irrational  numbers. 

227.  A  rational  arithmetical  number  is  called  a  Commensurable 
Number,  and  an  irrational  arithmetical  number  is  called  an  Incom- 
mensurable Number. 

2,  f,  .54,  and  ,666  are  commensurable,  but  V2  is  incommensurable. 

In  algebra,  commensurable  and  incommensurable  numbers  may 
be  either  positive  or  negative. 

The  terms  rational  and  irrational  applied  to  algebraic  numbers 
relate  to  their  forms,  while  the  terms  commensurable  and  incom- 
mensurable relate  to  their  arithmetical  values. 

3  a,  a  +  &,  aj  —  3,  x^,  are  rational  but  not  necessarily  commensurable. 
For  a  may  represent  \/2,  b  may  represent  V5,  etc.  Again,  Vx  is  irrational, 
but  if  X  =  16,  Vx  is  commensurable. 

Incommensurable  numbers  obey  the  Commutative,  Associative,  and  Dis- 
tributive Laws,  but  the  proof  is  too  complicated  to  be  given  here. 


EVOLUTION  223 

228.  Since  the  nth  power  of  a  number  is  the  product  of  n  equal 
factors,  and  since  one  of  these  factors  is  a  root  of  the  power,  it 
follows  that  {\/ay'=a,  and  if  the  principal  root  is  meant,  Va"=a, 
when  n  is  a  positive  integer. 

In  the  statement  of  the  following  principles  and  in  Ax.  7  the 
term  root  means  principal  root. 

229.  Principles.  — 1.  Law  of  Signs.  —  An  odd  root  of  a  number 
has  the  same  sign  as  the  number  ;  an  even  root  of  a  positive  number 
is  positive  ;  an  even  root  of  a  negative  number  is  impossible,  or 
imaginary. 

2.  Law  of  Exponents.  —  TJie  exponent  of  any  root  of  a  number  is 
equal  to  the  exponent  of  the  given  number  divided  by  the  index  of  the 
root. 

3.  Any  root  of  a  product  is  equal  to  the  product  of  that  root  of 
each  of  the  factors. 

4.  Any  root  of  the  quotient  of  two  numbers  is  equal  to  the  quotient 
of  that  root  of  each  of  the  numbers. 

Even  roots  of  negative  numbers  will  be  discussed  later. 

The  above  principles  may  be  established  as  follows : 

Principle  1  follows  from  the  Law  of  Signs  for  multiplication. 

Principle  2.     When  m  and  n  are  positive  integers, 
§  218,  Prin.  2,  a««  =  (a'")". 

Taking  the  nth  root,  §  228,  y/a^'  =a"» ; 

.-.  y/a^  =  a™""^  =  a"». 

Thus,  Va^  =  a\ 

Principle  3.     When  n  is  a  positive  integer, 
§  228,  §  218,  Prin.  3,  ah  =  (  VaY  x  (  '</hY  =  (  ^a  x  ^ft)«. 

Taking  the  nth  root,  y/ab  =  v^a  x  ^b. 

Thus,  v/27^  =  ^27  X  \/^  =  3  a« 

Principle  4.     When  w  is  a  positive  integer, 


§228,  §218,  Prin.  4, 


b     {VbY     WbJ  ' 


Taking  the  nth  root,  .;*/-= — • 

\b     Vb 


224  ACADEMIC  ALGEBRA 

230.    Evolution  of  monomials. 

Examples 

1.  What  is  the  square  root  of  36  a%^  ? 

Solution. — Since,  in  squaring  a  monomial,  §  218,  the  coefficient  is 
squared  and  the  exponents  of  the  letters  are  multiplied  by  2,  to  extract  the 
square  root,  the  square  root  of  the  coefficient  must  be  found,  and  to  it  must 
be  annexed  the  letters  each  with  its  exponent  divided  by  2, 

The  square  root  of  36  is  6,  and  the  square  root  of  the  literal  factors  is  a%. 
Therefore,  the  principal  square  root  of  36  a^h'^  is  6  a^b. 

The  square  root  may  also  be  —  6  a^b,  since  -  6  a^^  x  —  6  a^^  =  36  a^b'^. 

.-.  v'36  a^b-^  =  ±  6  a^b. 

2.  What  is  the  cube  root  of  -  125  xhf^  ? 

Solution.  V-  125  x^i  =  -  ^  ^V- 

To  find  the  root  of  an  integral  term : 

EuLE.  —  Extract  the  required  root  of  the  numerical  coefficient, 
aymex  to  it  the  letters  each  with  its  exponent  divided  by  the  index  of 
ihe  root  sought,  and  prefix  the  proper  sign  to  the  result. 

To  find  the  root  of  a  fractional  term : 

Rule.  —  Find  the  required  root  of  both  numerator  and  denomi- 
nator and  prefix  the  proper  sign  to  the  resulting  fraction. 

Find  the  indicated  root : 


3. 

</a%''&'. 

4. 

^a%^'c'\ 

5. 
6. 

i/aVf'. 

7. 

■Vx'"fz'^. 

8. 

i/-Sa%''. 

9. 

-\/-32ajV"- 

10. 

V16  xy. 

11. 

^-a'Wx'\ 

12. 

^-243/«. 

13. 

-v/16  m'7i\ 

14 


15. 


16. 


17. 


18. 


19. 


V  128  a^^* 


4 


32  aV 


24:3  f 


256  a^ 
6561* 


3/     125 

4 


x'Y' 


1728  c3 


^4ny2n 


nlb^'^c^a^'' 

\  On  ^n.j&n 


2«^n^n 


EVOLUTION  225 

231.    To  extract  the  square  root  of  a  polynomial. 

1.  Since  a^  +  2ab-\-b^  is  the  square  of  {a-\-l)),  what  is  the 
square  root  of  a^  +  2  a6  +  6^  ? 

2.  How  may  the  first  term  of  the  square  root  be  found  from 

3.'  How  may  the  second  term  of  the  square  root  be  found  from 
2  ah  J  the  second  term  of  the  power  ? 

4.  What  are  the  factors  of  a^  +  2  a&  +  6^  ? 

5.  Since  2ab  -{-W  is  equal  to  6(2  a  +  6),  what  are  the  factors 
of  the  last  two  terms  of  the  square  of  a  binomial  ? 

6.  By  what  divisor,  then,  must  the  last  two  terms  of 
a^ -]- 2  ah -\- h^  be  divided  sb  that  the  quotient  may  be  the  second 
term  of  the  square  root  ? 

Examples 

1.  Find  thd  process  for  extracting  the  square  root  of 
a2  +  2  a6  +  h\ 

PROCESS 

a^-\-2ah  +  l^\  a  +  h 
a^ 

Trial  divisor,  2a  2oh-{-h^ 

Complete  divisor,  2a-\-h  2ah  -\-h^ 


Explanation.  — Since  a^  ■^2ab  +  b^  is  the  square  of  (a  +  6),  we  know 
that  the  square  root  of  a^  -\-  2  ab  -\-  b^  is  o  +  6. 

Since  the  first  term  of  the  root  is  a,  it  may  be  found  by  taking  the  square 
root  of  a^,  the  first  term  of  the  power.  Subtracting  a^,  there  is  a  remainder 
of  2  a&  +  b"^. 

The  second  term  of  the  root  is  known  to  be  6,  and  that  may  be  found  by 
dividing  the  first  term  of  the  remainder  by  twice  the  part  of  the  root  already 
found.     This  divisor  is  called  a  trial  divisor. 

Since  2ab  -{■  b^  is  equal  to  b(2  a  -\-  6),  the  complete  divisor  which  multi- 
plied by  b  produces  the  remainder  2  ab  -}■  b'^  is  2  a  +  6  ;  that  is,  the  complete 
divisor  is  found  by  adding  the  second  term  of  the  root  to  twice  the  root 
already  found. 

Multiplying  the  complete  divisor  by  the  second  term  of  the  root  and  sub- 
tracting, there  is  no  remainder,  consequently,  a  +  6  is  the  required  root. 

▲CAD.  ALG.  — 15 


226.  ACADEMIC  ALGEBRA 

2.    Find  the  square  root  of  9x^  —  SO  xy  -^25  y^. 

PROCESS 

Qx'-SOxy-h  25  /  \Sx-5y 
9  or 

6a; 

6  X  —  5y 


30  xy  +  25  y^ 
30  xy  +  25  ^2 


Since,  in  squaring  a-{-h -\- c^a-\-h  may  be  represented  by  a?,  and 
the  square  of  the  number  by  ic^  +  2  icc  +  c^,  it  is  obvious  that  the 
square  root  of  a  number  whose  root  consists  of  more  than  two 
terms  may  be  extracted  in  the  same  way  as  in  Example  1,  by  con- 
sidering the  terms  already  found  as  one  term. 

3.    Find  the  square  root  of  4  o;^  + 12  a^  —  3  it-^  —  18  a;  +  9. 

PROCESS 

4a;^  +  12a^-    3a^-18aj  +  9  |  2  a;^  +  3  a;  -  3 
4.0? 


4  a;2  +  3  a; 


4  a;^  -|-  6  a? 
4a^  +  6a;-3 


12a;3-    Sx" 

12a;^+    9a;^ 

-  12  a;2  _  13  ^  _^  9 


-12ar^-18a;-f  9 


Explanation. — Proceeding  as  in  the  previous  example,  the  first  two 
terms  of  the  root  are  found  to  be  2x^  +  '6  x. 

Considering  (2  ofi  +  3  a;)  as  the  first  term  of  the  root,  the  next  term  of  the 
root  is  found  as  the  second  term  of  a  root  is  found,  by  dividing  the  remainder 
by  twice  the  part  of  the  root  already  found.  Hence,  the  trial  divisor  is 
4  x2  4-  6  x,  and  the  next  term  of  the  root  is  —  3.  Annexing  this,  as  before, 
to  the  trial  divisor  already  found,  the  entire  divisor  is  2  x'^  +  3  ic  —  3.  This 
multiplied  by  —  3  and  the  product  subtracted  from  —  12  x^  —  18  a;  +  9,  leaves 
no  remainder.     Hence,  the  square  root  of  the  number  is  2  x^  +  3  x  —  3. 

Rule.  —  Arrange  the  terms  of  the  polynomial  with  reference  to 
the  consecutive  powers  of  some  letter. 

Extract  the  square  root  of  the  first  term,  write  the  result  as  the 
first  term  of  the  root,  and^  subtract  its  square  from  the  given 
polynomial. 


EVOLUTION  227 

Divide  the  first  term  of  the  remainder  by  twice  the  root  already 
found,  as  a  trial  divisor,  and  the  quotient  will  he  the  next  term  of 
the  root.  Write  this  result  in  the  root,  and  annex  it  to  the  trial 
divisor  to  form  the  complete  divisor. 

Multiply  the  complete  divisor  by  this  term  of  the  root,  and  sub- 
tract the  product  from  the  first  remainder. 

Find  the  next  term  of  the  root  by  dividing  the  first  term  of  the  re- 
mainder by  the  first  term  of  the  trial  divisor. 

Form  the  complete  divisor  as  before  and  continue  in  this  manner 
until  all  the  terms  of  the  root  are  found. 

Find  the  square  root  of  the  following  : 

4.  4a;2H-12a;  +  9.  7.   a^  +  an/H-Jy^. 

5.  4a^-f  20a;  +  25.  8.    4.0^  -  62x-\-im. 

6.  25ar2  +  40a;  +  16.  v,  9.    (a  +  6)2_  4(a  +  6) -f- 4 

10.  a;®  +  4  ic^  +  2  a;*  +  9  a;2  —  4  a;  -h  4. 

11.  ^a^-12a^^l0x'-i.x-^l. 

12.  a^-6a^y  +  13a^2/2_i2an/3  +  4y*. 

13.  a^  +  2aV-aV-2a2a^H-a». 

14.  25a;*  +  4-12a;-30a:3-f29ar^. 

15.  l  +  6a;  +  15a^  +  20a^H-15a;*  +  6ar'-fa;«. 

16.  l-2a;  +  3.T2-4ar^  +  3a;^-2ar^  +  a:«. 

17.  a^-2a26  +  2a2c2-25c2  +  62  +  c*. 

18.  4a2-12a6-hl6ac  +  962  +  16c2-246c. 

19.  9a;2_^252/2  +  922_30a;2/4-18a^-302/2!. 

^.  20.    ic2  +  2a;-l-?  +  i. 
X     ar 

.   .X.   x'  +  a^  +  ^  +  l  +  i. 
23.   a!»  +  4a!' -  2a!«  -  20a!=  -  3a;*  +  32k»  +  4*' -  16a!  +  4. 


228  ACADEMIC  ALGEBRA 

24.   Find  four  terms  of  the  square  root  of  1  +  a;. 


So 


LOTION 


1 


2  +  ix 


X 


2  +  x-^x'^ 


-    ix^-i^s  +  Ax't 


2  +  x-lx^-hjhx^  I  1x3-^^x4 
Find  the  square  root  of  the  following  to  four  terms : 

25.  1-a.  27.    a^-1.  29.    /  +  3. 

26.  a^  +  l.  28.   4 -a.  30.    a- +  2  &. 

SQUARE  ROOT  OF  ARITHMETICAL  NUMBERS 

12  =  1  10^  =  100  1002  =  10000 

92  =  81  992  =  9801  9992  =  998001 

232.  1.  How  many  figures  are  required  to  express  the  square 
of  a  number  expressed  by  1  figure  ?  2  figures  ?  3  figures  ?  4 
figures  ? 

2.  How  does  the  number  of  figures  in  the  square  of  a  number 
compare  with  the  number  of  figures  in  the  number  ? 

3.  How  many  figures  are  there  in  the  square  root  of  a  number 
that  is  expressed  by  4  figures  ?  by  3  figures  ?  by  5  figures  ? 
by  6  figures  ?   by  7  figures  ? 

4.  How,  then,  may  the  number  of  figures  in  the  square  root  of 
a  given  number  be  found  ? 

Principles.  —  1.  The  square  of  a  number  is  expressed  by  twice 
as  many  figures  as  there  are  in  the  number  itself,  or  by  one  less  than 
twice  as  many. 

2.  Tlie  orders  of  units  in  the  square  root  of  a  number  correspond 
to  the  number  of  periods  of  two  figures  each  into  ivhich  the  number 
can  be  separated,  beginning  at  U7iits. 


EVOLUTION  229 

233.  If  the  number  of  units  expressed  by  the  tens'  digit  is 
represented  by  t  and  the  number  of  units  expressed  by  the  units' 
digit  by  u,  the  square  of  a  number  consisting  of  tens  and  units 
will  be  represented  by  (t  +  uY,  or  f  -\-2tu-\-  iii 

Thus,  25  =  2  tens  +  5  units,  or  20  +  5  units, 

and  252  =  202  +  2 (20  X  5)  +  52  =  625. 

Examples 
1.    What  is  tlie  square  root  of  2809  ? 

Explanation.  —  Accordinoj  to  Prin. 
FIRST    PROCESS  «      c  00,     .,  ^  r  f     •       *u 

2,    §  231,  the  orders  of  units  m  the 

28 '09  I  50  +  3         square  root  of  a  number  may  be  deter- 

t^  =  25  00  '         mined  by  separating  the  number  into 

~ri  periods  of  two  figures  each,  beginning 

at  units.     Separating  2809  thus,  there 

are  found  to  be  two  orders  of  units  in 

the  root;  that  is,  it  is  composed  of 

tens  and  units. 


2^  =  100 
u=     3 


2t  +  u  =  103 


3  09 

Since  the  square  of  tens  is  hundreds. 


and  the  hundreds  of  the  power  are  less  than  36,  or  6^,  and  more  than  25, 
or  52,  the  tens'  figure  of  the  root  must  be  5.  5  tens,  or  50,  squared  is  2500, 
and  2500  subtracted  from  2809  leaves  309,  which  is  equal  to  2  times  the 
tens  X  the  units  -f  the  units2. 

Since  two  times  the  tens  multiplied  by  the  units  is  much  greater  than  the 
square  of  the  units,  309  is  a  little  more  than  2  times  the  tens  multiplied  by 
the  units.  Therefore,  if  309  is  divided  by  2  times  the  tens,  or  100,  the  trial 
divisor,  the  units  are  found  to  be  3.  And  since  the  complete  divisor  is  found 
by  adding  the  units  to  twice  the  tens,  the  complete  divisor  is  100  +  3,  or  103. 
This  multiplied  by  3  gives  as  a  product  309,  which  subtracted  from  309  leaves 
no  remainder.     Therefore,  the  square  root  of  2809  is  53. 

SECOND    PROCESS 

28*09  I  53  Explanation.  —  In  practice  it  is  usual  to 

<*  =  25  place  the  figures  of  the  same  order  in  a  col- 

3Q9  umn,  and  to  disregard  the  ciphers  on  the 

right  of  the  products. 


2^  =  100 

u=     3 

2t  +  u  =  103 


309 


Since  any  number  may  be  regarded  as  composed  of  tens  and 
units,  the  processes  given  above  have  a  general  application. 

Thus,  346  =  34  tens  +  6  units ;  2377  =  237  tens  +  7  units. 


230 


ACADEMIC  ALGEBRA 


2.   Find  the  square  root  of  104976. 


Solution 


Trial  divisor  =  2  x  30  =60 

Complete  divisor  =  60  +  2  =62 
Trial  divisor         =  2  x  320  =  640 
Complete  divisor  =  640  +  4  =  644 


10.49.76 

9 

149 

124 

25  76 

25  76 

324 


Rule.  —  Separate  the  number  into  periods  of  two  figures  each, 
beginning  at  units.  • 

Find  the  greatest  square  in  the  left-hand  period  and  ivrite  its  root 
for  the  first  figure  of  the  required  root. 

Square  this  root,  subtract  the  result  from  the  left-hand  period,  and 
annex  to  the  remainder  the  next  period  for  a  new  dividend. 

Double  the  root  already  found,  with  a  cipher  annexed,  for  a  trial 
divisor,  and  by  it  divide  the  dividend.  The  quotient  or  the  quotient 
diminished  will  be  the  second  figure  of  the  root.  Add  to  the  trial 
divisor  the  figure  last  found,  midtiply  this  complete  divisor  by  the 
figure  of  the  root  found,  subtract  the  product  from  the  dividend,  and 
to  the  remainder  annex  the  next  period  for  the  next  dividend. 

Proceed  in  this  manner  until  all  the  periods  have  been  used.  The 
result  will  be  the  square  root  sought. 

1.  When  the  number  is  not  a  perfect  square,  annex  periods  of  decimal 
ciphers  and  continue  the  process. 

2.  Decimals  are  pointed  off  from  the  decimal  point  toward  the  right. 

3.  The  square  root  of  a  fraction  may  he  found  by  extracting  the  square 
root  of  both  numerator  and  denominator  separately  or  by  reducing  it  to  a 
decimal  and  then  extracting  its  root. 


Extract  the  square  root  of  the  following : 
3.    529.  9.    57121. 


4.  2209. 

5.  4761. 

6.  7921. 

7.  17424. 

8.  19321. 


10.  42025. 

11.  95481. 

12.  186624. 

13.  165649. 

14.  134689. 


15.  125316. 

16.  455625. 

17.  992016. 

18.  2480.04. 

19.  10.9561. 

20.  .001225. 


EVOLUTION 

23 

21.    10201. 

24.    1332.25. 

27. 

540.5625. 

22.   95481. 

25.    101.0025. 

28. 

1.018081. 

23.    363609. 

26.    111.0916. 

29. 

13003236. 

30.  m. 

32.    iff.                   34. 

Mf. 

36.  m. 

31.  m- 

33.    3-VA.                35. 

'iU'U- 

37.  m. 

Extract  the 

square  root  to  four  decimal  places : 

. 

38.    f. 

40.    |.                        42. 

f 

44.    |. 

39.    f. 

41.    .6.                       43. 

f 

45.    A. 

234.   To  extract  the  cube  root  of  a  polynomial. 

1.  Since  a*  +  3  a^6  -f-  3  ab^  H-  b^  is  the  cube  of  (a  +  b),  what  is 
the  cube  root  of  a^  +  3  a^b  -^Sab^  +  b^? 

2.  How  may  the  first  term  of  the  root  be  found  from  a^  +  3  a^b 
-{-Sab^  +  b^? 

3.  How  may  the  second  term  of  the  root  be  found  from  the 
second  term  of  the  power,  3  a%  ? 

4.  What  are  the  factors  of  3  a^b  -{- S  ab^  +  b^  ? 

5.  Since  3  a^6  +  3  ab^  +  b^  is  equal  to  6 (3  a^  +  3  a6  +  b^,  by 
what  number  must  the  last  three  terms  of  a^  -f-  3  a^b  +  3  a6^  +  b^ 
be  divided  so  that  the  quotient  may  be  the  second  term  of  the 
cube  root? 

Examples 

1.  Find  the  process  for  extracting  the  cube  root  of  a^-{-Sa^b 
+  3ab^-{-b\ 

PROCESS 

a^-\-Sa^b-\-Sab^-\-b^  |  a-hb 
0? 

Trial  divisor,  3  a^ 

Complete  divisor,   3a^+Sab  -\-b^ 


3a^b-\-3ab^-^b^ 
3a26+3a&2+63 


Explanation.  — Since  a^  +  3  a'^b  -{■  S  ab"^  +  b^  is  the  cube  of  (a  +  b),  we 
know  that  the  cube  root  of  a^  +  3  a'^b  +  .3  ab^  +  b^  is  a  -}-  b. 

Since  the  first  term  of  the  root  is  a,  it  may  be  found  by  taking  the  cube 
root  of  a^,  the  first  term  of  the  power.  Subtracting,  there  is  a  remainder  of 
3  «26  +  3  ab^  +  b^ 


282  ACADEMIC  ALGEBRA 

The  second  term  of  the  root  is  known  to  be  &,  and  that  may  be  found  by 
dividing  the  first  term  of  the  remainder  by  3  times  the  square  of  the  part  of 
the  root  already  found.     This  divisor  is  called  a  trial  divisor. 

Since  3  a^ft  +  3  ah"^  +  6^  is  equal  to  h  {^  a^  +  Z  ah  -\-  62),  the  complete  divi- 
sor, which  multipUed  by  6  produces  the  remainder  3  a%  +  3  ah"^  +  6',  is 
3  a^  4-  3  a5  +  62 .  that  is,  the  complete  divisor  is  found  by  adding  to  the 
trial  divisor  3  times  the  product  of  the  first  and  second  terms  of  the  root  and 
the  square  of  the  second  term  of  the  root. 

Multiplying  the  complete  divisor  by  the  second  term  of  the  root,  and  sub- 
tracting,'there  is  no  remainder  ;  consequently,  a  +  6  is  the  required  root. 

Since,  in  cubing  a  +  6-f-c,  a  +  h  may  be  expressed  by  x,  its 
cube  will  be  £c3  +  3  a;^c  +  3  icc^  -t-  c^.  Hence,  it  is  obvious  that  the 
cube  root  of  an  expression  whose  root  consists  of  more  than  two 
terms  may  be  extracted  in  the  same  way  as  in  example  1,  by  con- 
sidering the  terms  already  found  as  one  term. 

2.   Find  the  cube  root  of  6«  -  3  6^  +  5  6^  -  3  6  -  1. 


3  6* 
3  6*- 

PROCESS 

6« 

_1  162-6- 

Trial  divisor, 
Complete  divisor, 

-3  63+6^ 

-36«+568 

_355_^3  54_53 

Trial  divisor, 
Complete  divisor. 

3  6*- 
36*- 

-6  63+3  6 
-6  63+36 

2       |_3  6*+6  63- 
+  1    -3  6*+6  63- 

-36-1 
-3  6-1 

Explanation.  —  The  first  two  terms  are  found  in  the  same  manner  as  in 
the  previous  example.  In  finding  the  next  term,  6^  _  ft  is  considered  as  one 
term,  which  we  square  and  multiply  by  3  for  a  trial  divisor.  Dividing  the 
remainder  by  this  trial  divisor,  the  next  term  of  the  root  is  found  to  be  —  1 . 
Adding  to  the  trial  divisor  3  times  (6^  _  ft)  multiplied  by  —  1,  and  the  square 
of  —  1,  we  obtain  the  complete  divisor.  This  multiplied  by  —  1,  and  the 
product  subtracted  from  — 3  6*  +  6ft3  —  3ft  —  1,  leaves  no  remainder.  Hence, 
the  cube  root  of  the  polynomial  is  ft^  —  ft  —  1. 

Rule.  —  Arrange  the  polynomial  with  reference  to  the  consecutive 
powers  of  some  letter. 

Extract  the  cube  root  of  the  first  term,  write  the  result  as  the  first 
term  of  the  root,  and  subtract  its  cube  froyn  the  given  polynomial. 

Divide  the  first  term  of  the  remainder  by  3  times  the  square  of 
the  root  already  found,  as  a  trial  divisor,  and  the  quotient  will  be 
the  next  term  of  the  root. 


EVOLUTION  233 

Add  to  this  trial  divisor  3  times  the  product  of  the  first  and  second 
terms  of  the  root,  and  the  square  of  the  second  term.  TJie  result 
will  he  the  complete  divisor. 

Multiply  the  complete  divisor  by  the  last  term  of  the  root  found, 
and  subtract  this  product  from  the  dividend. 

Find  the  next  term  of'  the  root  by  dividing  the  first  term  of  the 
remainder  by  the  first  term  of  the  trial  divisor. 

Form  the  complete  divisor  as  before,  and  continue  m  this  manner 
until  all  the  terms  of  the  root  are  found. 

Find  the  cube  root  of 

3.  oc^  —  3  xhj -{- 3  xy^  —  y^. 

4.  m3-9  7/i2  +  27m-27. 

5.  8  m^  -  60  m-zi  +  150  m?r  -  125  w^ 

6.  27  a»  -  189  x'y  ■+■  441  xy^  -  343  f. 

7.  125  a«  +  675  a^a; -h  1215  aar^H- 729  053. 

8.  1000 p''- 300 p*q  + 30 pY-^' 

9.  m«  +  6  m^  4- 15  m*  +  20  m^  +  15  m^  4-  6  mH- 1. 

10.  x^-6x'-\-15x*-20a^-\-15a^-6x-\-l. 

11.  a.-^  +  3  ar'H- 9  a;* +  13  0^  +  18^5^  +  12  a; +  8. 

12.  ««+  12  ar^  +  63  a;*  +  184  a;3  _^  315  aj2  ^  300  x  +  125. 

13.  x^-\-6a^-  18  x*  -  1000  +  180  ar^  -  112  a^  +  600  x. 

14.  a:3_i2a^  +  54a;-112  +  i5§_^  +  l 

X        or      of 

15.  1  -  6  a  +  21  a2  _  44  a3  ^  63  a*  -  54  a^  +  27  a«. 

16.  64  -  144p  +  156 p2  _  99^58  ^  39^4  _  9^5  ^^e 
j^     aW_c^     3aca;^     3  a^bx^ 


c^  b^  b  c 

x'  '  ~^  '  x*'^a^ 


18.   jc«+15a;2  +  i|  +  20  +  -^  +  i  +  6a;^. 


234  ACADEMIC  ALGEBRA 

CUBE  ROOT  OP  ARITHMETICAL  NUMBERS 
13  =  1.  103  =  1000.  1003  ^  1000000. 

33  =  27.  303  =  27000.  3003  =  27000000. 

93  =  729.  993  =  970299.  9993  ^  997002999. 

• 

235.  1.  How  many  figures  are  there  in  the  cubes  of  numbers 
that  have  1  figure  ?    2  figures  ?    3  figures  ?   4  figures  ? 

2.  What  places,  then,  belong  to  the  cube  of  units  ?  of  tens  ? 
of  hundreds  ?   of  thousands  ? 

3.  How  many  figures  are  there  in  the  cube  root  of  a  number 
expressed  by  4  figures  ?   5  figures  ?   6  figures  ?    7  figures  ? 

4.  How  may  the  number  of  figures  in  the  cube  root  of  a 
number  be  found?  ^ 

Principles.  —  1.  The  cube  of  a  number  is  expressed  by  three 
times  as  many  figures  as  the  number  itself,  or  by  one  or  two  less 
than  three  times  as  many. 

2.  The  orders  of  units  in  the  cube  root  of  a  number  correspond  to 
the  number  of  periods  of  three  figures  each,  into  which  the  number 
can  be  separated,  beginning  at  units. 

236.  If  the  number  of  units  expressed  by  the  tens'  digit  is 
represented  by  t,  and  the  number  of  units  expressed  by  the  units' 
digit  by  u,  the  cube  of  a  number  consisting  of  tens  and  units  will 
be  represented  by  {t  +  uy,  or  ^  +  3  fu  -\-^tu^  -\-  u^. 

Thus,  25  =  2  tens  +  5  units,  or  20  +  5  units, 

and  253  =  203  +  3  (202  x  5)  +  3  (20  x  52)  +  53  =  16625. 

Examples 
1.    What  is  the  cube  root  of  12167  ? 

FIRST    PROCESS 

12.167  I  20  +  3 


^  =  8  000 

Trial  divisor,  ^f  =  1200 


Stu=    180 
u^=       9" 


Complete  divisor,  =  1389 


4  167 


4  167 


EVOLUTION 


235 


Explanation.  —  By  Prin.  2,  §  235,  the  orders  of  units  may  be  determined 
by  separating  the  number  into  periods  of  three  figures  each,  beginning  at 
units.  Separating  12167  thus,  there  are  found  to  be  two  orders  of  units  in 
the  root ;  that  is,  the  root  is  composed  of  tens  and  units. 

Since  the  cube  of  tens  is  thousands,  and  the  thousands  of  the  power  are 
less  than  27,  or  3^,  and  more  tlian  8,  or  2^,  the  tens'  figure  of  the  root  is  2. 
2  tens,  or  20,  cubed  is  8000,  and  8000  subtracted  from  12167  leaves  4167, 
which  is  equal  to  3  times  the  tens2  x  the  units  +  3  times  the  tens  x  the  units2 
+  the  units^. 

Since  3  times  the  tens2  x  the  units  is  much  greater  than  3  times  the  tens 
X  the  units2,  4167  is  a  little  more  than  3  times  the  tens2  x  the  units.  If, 
then,  4167  is  divided  by  3  times  the  tens2,  or  by  1200,  the  trial  divisor,  the 
quotient  3  will  be  the  units  of  the  root,  provided  proper  allowance  has  been 
made  for  the  additions  necessary  to  obtain  the  complete  divisor. 

Since  the  complete  divisor  is  found  by  adding  to  3  times  the  tens2  3  times 
the  tens  x  the  units  and  the  units2,  the  complete  divisor  is  1200  +  180  +  9,  or 
1389.  This  multiplied  by  3  gives  4167,  which  subtracted  from  4167  leaves  no 
remainder,     'llierefore,  the  cube  root  of  12167  is  20  +  3,  or  23. 


SECOND    PROCESS 

12.167  [ 
^  =  8 

3  «2  =  1200 
Stu=    180 


23 


1389 


4  167 


Explanation.  —  In  practice  it  is  usual  to 
place  figures  of  the  same  order  in  a  column, 
and  to  disregard  the  ciphers  on  the  right  of  the 
products. 


4  167 


2.    What  is  the  cube  root  of  1740992427  ? 

Solution 


•  3  «2  =  3(10)2 
3«t<  =  3(10  X  2) 
m2  =  22 

= 

300' 

60 

4 

1.740.992.427  [J203 
1 
740 

3(2  =3(120)2 

364 
4320 

0 

728 
12992 

%\ 

3«2  =3(1200)2 
3  tu  =  3(1200  X 
w2  =  32 

3)  = 

4320000 

10800 

9 

12992427 

433080 

9 

12992427 

Since  a  root  expressed  by  any  number  of  figures  may  be  regarded  as  com- 
posed of  tens  and  units,  the  processes  of  example  1  have  a  general  application. 
TSu!^    V20  =  12  tens  +  0  units  ;  and  1203  =  120  tens  +  3  units. 


236 


ACADEMIC  ALGEBRA 


Since  the  third  figure  of  the  root  is  0,  it  is  not  necessary  to  form  the  com. 
plete  divisor,  inasmuch  as  the  product  to  be  subtracted  will  be  0. 

Rule. — Separate  the  numbers  into  periods  of  three  figures  each, 
beginning  at  units. 

Find  the  greatest  cube  in  the  left-hand  period,  and  write  its  root 
for  the  first  term  of  the  required  root.  Cube  the  root,  subtract  the 
result  from  the  left-hand  period,  and  annex  to  the  remainder  the  next 
period  for  a  new  divideiid. 

Take  3  times  the  square  of  the  root  already  found,  with  two  ciphers 
annexed,  for  a  trial  divisor,  and  by  it  divide  the  dividend.  The 
quotient,  or  quotient  diminished,  will  be  the  second  figure  of  the  root. 

To  this  trial  divisor  add  3  times  the  product  of  the  first  part  of 
the  root  with  a  cipher  annexed,  multiplied  by  the  second  part,  and 
also  the  square  of  the  second  part.  Their  sum  will  be  the  complete 
divisor. 

Multiply  the  complete  divisor  by  the  second  part  of  the  root,  and 
subtract  the  product  from  the  dividend. 

Continue  thus  until  all  the  figures  of  the  root  have  been  found. 

1.  When  there  is  a  remainder  after  subtracting  the  last  product,  annex 
decimal  ciphers,  and  continue  the  process. 

2.  Decimals  are  pointed  off  from  the  decimal  point  toward  the  right. 

3.  The  cube  root  of  a  common  fraction  may  be  found  by  extracting  the 
cube  root  of  both  numerator  and  denominator  separately,  or  by  reducing  it 
to  a  decimal  and  then  extracting  its  root. 

Extract  the  cube  root  of 


3.    29791. 

9.   2406104. 

15. 

.000024389. 

4.   54872. 

10.   69426531. 

16. 

.001906624. 

5.   110592. 

11.   28372625. 

17. 

.000912673. 

6.   300763. 

12.   48.228544. 

18. 

.259694072. 

7.   681472. 

13.   17173.512. 

19. 

926.859375. 

8.   941192. 

14.   95.443993. 

20. 

514500.058197. 

Extract  the  cube  root  to  three  decimal  places : 

21.    2. 

23.    .8. 

25. 

A- 

27.    f 

22.    6. 

24.    .16. 

26. 

i- 

28.    ^. 

EVOLUTION  237 

237.    To  extract  any  root  of  a  polynomial. 

To  find  a  formula  for  obtaining  the  complete  divisor  in  extract- 
ing the  fourth,  fifth,  sixth,  or  any  required  root  of  a  polynomial, 
raise  (a  +  h)  to  the  required  power,  and  separate  all  the  terms 
after  the  first  into  two  factors  one  of  which  shall  be  the  second 
term  of  the  root.  The  other  factor  will  be  the  formula  for  the 
complete  divisor. 

(a  +  6)5  =  ^5  +  5  Qj45  ^  10  a^h"^  +  10  a^h^  +  bah^-\-  b^ 

Trial  divisor,  5  a*. 

Complete  divisor,  5  a*  +  10  a^b  +  10  a%^  +  5ab^+  &*. 

(a  +  6)7  =  a^  +  7  a^ft  +  21  a^b'^  +  35  a*&8  +  35  a^b^  +  21  a^b^  +  7  a6«  +  6^. 

Trial  divisor,  7  a^. 

Complete  divisor,  7  a^  +  21  a^ft  +  35  a^b-  +  35  a^b^  +  21  a^b^  +  7  ab^ -\-  6«. 

Since  the  fourth  power  is  the  square  of  the  second  power,  the 
sixth  power  the  cube  of  the  second  power,  etc.,  any  root  whose 
index  is  4,  6,  8,  9,  etc.,  may  be  found  by  extracting  successively 
the  roots  corresponding  to  the  factors  of  the  index. 

The  fourth  root  may  be  obtained  by  extracting  the  square  root  of  the 
square  root ;  the  sixth  root,  by  extracting  the  cube  root  of  the  square  root, 
or  the  square  root  of  the  cube  root ;  the  eighth  root,  by  extracting  the  square 
root  of  the  square  root  of  the  square  root. 

Examples 

1.  Find  the  fourth  root  of  16  -  32  x  +  24  a^  -  8  x^  +  x\ 

2.  Find  the  fourth  root  of  a^  + 12  a^2/  +  54  a.-^/  -f  108  xf  +  81  y\ 

3.  Find  the  fourth  root  of  16  m^  -  32  m^  -f  24  m^  -  8  m  +  1. 

4.  Find  the  fifth  root  of  32:^5  +  80  a;*  +  80  a:3_|_  40  ar^  4.10  a;  +  1. 

5.  Find  the  fifth  root  of  ai«+15a«+90a«4-270a*4-405a24-243. 

6.  Find  the  sixth  root  of  a;«  -  12  ar'  -f  60  a;*  -  160  a;^  +  240  x" 

-192a;  +  64. 

7.  Find  the  sixth  root  of  64  .'r«  -  576  a;«  +  2160  x'  -  4320  0? 

+  4860  a;2- 2916  .T  + 729. 

8.  Find  the  sixth  root  of  a;^  +  6  acxP  +  15  ah'^x'^  +  20  «Var« 

+  15  a*cV  + 6  aVa;-f-a«c«. 


238 


ACADEMIC  ALGEBRA 


238.    To  extract  any  root  of  an  arithmetical  number. 

Examples 
1.   Find  the  cube  root  of  42875. 

Solution 
By  factoring,  42875  =  5x5x5x7x7x7. 

.-.  §§  26,  229,  Prin.  3,     \/42875  =  5  x  7  =  35. 

Find,  by  factoring,  the  roots  indicated : 
4.    ^531441. 


2.  a/3375. 

3.  -v'i296.  5.    \/759375. 
8..  Find  the  sixth  root  of  1771561. 

Solution.  —The  square  root  of  1771561  is  1331. 

The  cube  root  of  1331  is  11. 
§  237,  \/l771561  =  11. 

Find  the  roots  indicated : 
9.    \/50625.  11.    ^531441. 


6.  V4084101. 

7.  -^'262144. 


13.    V24137569. 


10.    V46656. 


12.    -\/5764801. 


14.    V10604499373. 


239.   Factoring  by  evolution. 

Examples 

1.  Factor  x'^ -{-4.a^ -\-Sx^  +  Sx  — 5. 

Solution.  —  Extracting  the  square  root  as  far  as  possible,  the  root  ob- 
tained is  a;2  -f  2  X  +  2  with  a  remainder  of  —  9. 

Therefore,  x'^  +  4x^ -^^  Sx^  +  Sx- 5 

=  (a;2  +  2  iK  +  2)2  -  9 

§128,  =(x^  +  2x-{-2  +  S){x^  +  2x  +  2-S') 

=  (ic2  +  2  a: -f  5)(x'^  +  2  X  -  1). 

Factor  the  following  polynomials : 

2.  x^  +  6a^  +  lla^  +  6x-S. 

3.  ic«  +  2aj^  +  5a;*H-8a^  +  8a^  +  8a;4-3. 

4.  ir8-4a^  +  6aj^  +  6a^-19a^4-10a;  +  9. 

6.   4a^4-12iB^  +  25a;^  +  40a:»  +  40a^  +  32a;  +  16. 


THEOEY  OF  EXPONENTS 


240.  Thus  far  the  exponents  used  have  been  positive  integers 
only,  and  consequently  the  laws  relating  to  exponents  have  been 
obtained  in  the  following  restricted  forms  : 

1 .  a'"  X  a"  =  a"*"*""  when  m  and  n  are  positive  integers. 

2.  a"^ -i- ol"  =  a'^'"'  when  m  and  n  are  positive  integers  and 
m>  w. 

3.  (a"*)"  =  a"*"  when  m  and  n  are  positive  integers. 

4.  "C/a"*  =  a*""^"  when  ??i  and  ?i  are  positive  integers,  and  m  is  a 
multiple  of  n. 

5.  {ahy  =  a"6'*  when  m  is  a  positive  integer. 

If  all  restrictions  are  removed  from  m  and  n,  we  may  then  have 
expressions  like  a~^  and  a^.  But  such  expressions  are,  as  yet, 
unintelligible,  because  —  2  and  |  cannot  show  how  many  times  a 
number  is  used  as  a  factor. 

Since,  however,  these  forms  may. occur  in  algebraic  processes, 
it  is  important  to  discover  meanings  for  them  that  will  allow 
their  use  in  accordance  with  the  laws  already  established,  for 
otherwise  great  complexity  and  confusion  would  arise  in  the  pro- 
cesses involving  them. 

Assuming  that  the  law  of  exponents  for  multiplication, 

a"^  xa''  =  a™+",  (I) 

is  true  for  all  values  of  m  and  n,  the  meanings  of  zero,  negative, 
and  fractional  exponents  may  be  readily  discovered. 

Then,  having  verified  the  remaining  laws  of  exponents  for  these 
exponents,  all  the  laws  will  have  been  shown  to  be  of  general 
application  for  all  commensurable  exponents. 

239 


240  ACADEMIC  ALGEBRA 

241.   The  meaning  of  a  zero  exponent. 

1.  What  is  the  exponent  of  a  in  the  product  of  a^  xa^?  of 
a^  X  a}  ?  of  a^  x  a^,  if  the  law  of  exponents  for  multiplication  is 
true  for  all  values  of  m  and  n  ? 

2.  Since  a^xaP  =  a%  what  is  the  value  of  aP?  What  is  the 
value  of  a^?    of  6«?    of  ./?    of  m^? 

3.  What,  then,  is  the  value  of  the  zero  power  of  any  number  ? 

'  242.    Principle.  —  Any  number  with  a  zero  exponent  is  equal 
to  1. 

The  above  principle  may  be  established  as  follows : 

It  is  to  be  proved  that  a^  =  1. 

Since  Law  I  is  to  be  true  for  all  values  of  m  and  n,  §  240,  when  w  =  0, 

««  X  a^  =  a'^+o  =  a"*. 

Dividing  by  a«,  a'>  =  —  =  l. 

a"* 

243.   The  meaning  of  a  negative  exponent. 

1.  What  is  the  exponent  of  a  in  the  product  of  a*  x  a^?  of 
a*  X  a  ?  of  a*  X  a^  ?  of  a''  x  a~\  if  the  law  of  exponents  for  multi- 
plication is  true  for  all  values  of  m  and  n  ?   of  a^  x  a~^  ? 

2.  Since  a*  x  a~^  =  a^  and  since  a*  x  -  =  a%  to  what  fraction  is 

a 

a~^  equal  ?     Since  a'*  x  a~^  =  a%  to  what  fraction  is  a~^  equal  ? 

3.  What  is  another  expression  for  a~^  ?   for  x~^  ?   for  ?/-*  ? 

4.  What  is  the  equivalent  of  any  number  with  a  negative 
exponent  ? 

5 .  Since  a-^  =  i-,  to  what  is  a-^b  equal  ?     ^^— ^  ? 

a^  c 

6.  Since  — -  =  1  -f-  — ,  or  a^,  to  what  is  — -  x  -,  or  — -, equal  ? 

7.  Without  changing  the  value  of  the  fraction,  transfer  a  from 

a — ^h       .      (i^h 

the  numerator  to  the  denominator  in  ;    in  — :    from  the 

c  c 

denominator  to  the  numerator  in  ;    in  — 

a~h  ah 


THEORY  OF  EXPONENTS  241 

8.  How  may  a  factor  be  transferred  from  one  te.rm  of  a  frac- 
tion to  the  other  without  changing  the  value  of  the  fraction  ? 

244.  Principles.  —  1.  Any  iiumher  with  a  7iegative  exponent  is 
equal  to  the  reciprocal  of  the  same  number  with  a  numerically  equal 
positive  exponent. 

2.  Any  factor  may  be  transferred  from  one  term  of  a  fraction  to 
the  other,  without  changing  the  value  of  the  fraction,  if  the  sign  of 
the  eocponent  is  changed. 

The  above  principles  may  be  established  as  follows : 

Principle  1.    It  is  to  be  proved  that  a-"  =  — 

Since  §  240,  Law  I,  a^  x  W^  =  a^+'S  is  to  hold  for  all  values  of  m  and  w. 


when  m=—  n^ 

a- 

"x  a" 

_  a-*^+^  = 

ao; 

but,  §  242, 

ao 

=  1; 

.'.  Ax.  1, 

a 

«  X  a" 

=  1. 

Dividing  by  a**. 

a-" 

_  1 

Principle  2. 

It  is  to  be  proved 

that«I'"  = 
6-" 

-  K 

By  Prin.  1,  a 

-m  —  __ 

and 

6-n  = 

1 

1 
6"' 

Therefore, 

6-" 

~  1  "" 

a-      1 

6" 

Examples 
Write  the  following  with  negative  exponents : 

1.  l-na.  3.    1-f-a".  5.    c^a^x^. 

2.  l-!-al  4.    a-^Q^.  6.    am^-i-ba^. 

7.  Write  5a;~y  with  positive  exponents. 

Solution.  —  Prin.  1,        5  x~^y'^  =  5  y^—  =  ^-^. 

x^      x^ 

Write  the  following  with  positive  exponents : 

8.  2x-\  11.    a-^b-\  14.  Aa^c-^ 

9.  5a-'^.                       12.    x-^y-\                     15.  Sax'^ 
10.    3b-\                      13.    a-^bh-\                  16.  «"&-*• 

▲CAD.   ALG. 16 


242  ACADEMIC  ALGEBRA 


3  a^ 

17.  Write without  a  denominator. 

ar 

Solution.  —  Prin.  2,  ^  =  3  a^x-^ 

x^ 

Write  the  following  without  denominators : 

18.  — .  23.    -4^.  28.    f^ 


by  a~W  \y 

19.  WHt.  24.  ^.  29. 

20.  ^.  25.  3i^l  30. 

21.  ^.  36.  A.  31.  ?!'. 
6^n^  a;""*  b~^ 

22.  ~  27.  — .  32.  -^ 


y  (abf 

245.   The  meaning  of  a  fractional  exponent. 

1.  What  is  the  cube  root  of  a^?  How  is  its  exponent  ob- 
tained ?     Express  the  root  with  this  division  indicated. 

2.  In  a^  what  does  6  express  with  reference  to  a?  What 
does  3  express  with  reference  to  a^  ? 

3.  What  is  the  fifth  root  of  b^^?  Express  the  root  with  a 
fractional  exponent. 

4.  In  6~5-  what  does  15  express  with  reference  to  6  ?  What 
does  5  express  with  reference  to  b^^  ? 

5.  Express  the  cube  root  of  a  with  a  fractional  exponent ;  the 
fourth  root  of  a^ ;  the  third  root  of  the  seventh  powei'  of  a. 

6.  What  does  the  numerator  of  a  fractional  exponent  indi- 
cate ?     What,  the  denominator  ? 

7.  Since  a^  x  a^  X  a^  =  a^,  if  the  law  of  exponents  for  multi- 
plication holds  true  for  all  values  of  m  and  n,  of  what  is  a^  the 
cube  root  ? 

Since  a^  x  a^  X  a^  X  a^  x  a^  X  a^  =  a^,  of  what  is  a^  the  sixth 
power  ?     What  two  meanings  may  a^  have  ?   a^?   a*  ? 

8.  What  does  a  fractional  exponent  indicate  ? 


THEORY  OF  EXPONENTS  243 

246.    Principles. — 1.    The  numerator  of  a  positive  fractional 
exponent  indicates  a  p)ower  and  the  denominator  a  root. 

2.   A  positive  fractional  exponent  indicates  a  root  of  a  power  or  a 
power  of  a  root. 

Prin.  2  refers  to  principal  roots  only  (§  225). 

The  above  principles  may  be  established  as  follows : 

Let  p  and  q  be  any  positive  integers,  and  a  any  positive  number. 

—  p      p     p 
Since   —     =  -»   —  may  represent  any  positive  fraction. 

p 

1.  It  is  to  be  proved  that  in  a',  p  indicates  a  power  and  q  a  root. 
Since  Law  I,  or  a"*  x  a""  =  a'»+»,  is  to  hold  for  all  values  of  m  and  n, 

when  m  =  «  =  -, 

p        p        p_^_p        ^ 

P  P  P  ^  P  3p 

also  a^  X  a^  X  a^  =  ai  x  a'  =  a',  etc. 

P  P  P  OP 

Hence,        a^  x  a^  x  a^  •••  to  q  factors  =  a*  =  op. 
Taking  the  qth  root,  §  26  and  Ax.  7, 

p 

Z  ? 

Hence,  in  a^,  p  indicates  a  power  and  q  a  root,  and  a^  indicates  the  qth 

root  of  the  j?th  power  of  a. 

2.  It  is  to  be  proved  that  a^  =  Va^,  or  ( \/a)P. 

p         

By  the  previous  proof,  a*  =  Va'^.  (1) 

I 
If  p  =  1,  a»  =  Va. 

Raising  to  the  pth  power, 
1        1        1 
a^  X  a^  X  a^  '-'io  p  factors  =  (  •v'^)p, 

or,  Law  I,  a^  ={V~a)p.  (2) 


p 


Hence,  (1)  and  (2),  a^  indicates  the  qi\\  root  of  the  pih.  potoer  of  a,  or  the 

pth  power  of  the  qth  root  of  a. 

-^       1 
It  follows  from  §  244  that  a  *  =  — • 


244  ACADEMIC  ALGEBRA 

Examples 
1.   Express  -Va^bc~*  with  positive  fractional  exponents. 


Solution.  Va^bc^  =  a^b^c  ^  = 

Express  with  positive  fractional  exponents : 

2.  V^.  5.    (V^)^  8.  (\/^)-'. 

3.  V^y.  6.    {'\/yy-  9.  5^x~hj-\ 

4.  V^.  7.    {^ahy.  10.  2^(a-\-by. 

In  the  following,  large  numbers  may  be  avoided  by  extracting  the  root  first. 
Find  the  value  of 


11.  si 

15.    64i 

19. 

64-1 

12.   si 

16.   32i 

20. 

(~S)-i 

13.    8"*. 

17.    25l 

21. 

(-32)-t. 

14.    (-8)i 

18.    81^. 

22. 

«■*■.' 

Simplify : 

23.    -v^  +  a;^  +  8^  +  3a;^-5^- 

--^27^. 

24.    ^-y/^-^Bx'-Si 

»-^  +  2v'a;-i- 

-8^-2: 

xi 

25.    </^'-^/a^b-\--\ 

/a62  _  6  +  a  -f- 

■4:^a'b- 

-4« 

hi+</¥. 

Express  with  ra,rlical 

,  signs : 

26.    ai 

29.    ah\ 

32. 

ai^xK 

27.    x^. 

30.    i»%^. 

33. 

m^  -T-  n^. 

28.    x\ 

31.    a*6"*. 

34. 

X^  -f-  y^. 

247.  It  now  remains  to  complete  the  proof  that  the  other  laws 
of  exponents  are  of  general  application  for  commensurable  ex- 
ponents by  showing  that  they  apply  when  negative  and  fractional 
exponents  are  employed,  with  the  meanings  just  obtained. 


THEORY  OF  EXPONENTS  245 

248.  To  prove  that  the  law  of  exponents  for  division  is  general. 

It  is  to  be  proved  that  a^  -^  w*  =  a'"-"  for  all  values  of  m  and  n. 

Since  dividing  by  a"  is  equivalent  to  multiplying  by  its  reciprocal  — , 

a** 

§  106,  a""  ^  a»  =  a"»  X  — 

a" 

§  244,  Prin.  2,  =  a»«  x  a-» 

Hence,  for  all  values  of  m  and  n,   a"*  h-  a"  =  a"*-".  (II) 

249.  To  prove  that  the  law  of  exponents  for  involution  is  general. 

It  is  to  be  proved  that  («'")»  =  a""'  for  all  values  of  m  and  n. 
Case  1.  — Let  m  represent  any  value  and  n  a  positive  integer. 
Then,  (a*")"  =  a"*  x  «"*  x  a"*  •••  to  n  factors. 

Law  I,  =  a"»+"*+"'  -  ^  "  ^"" 

P 
Case  2.  —  Let  m  represent  any  value,  and  let  w  =  ^»  P  and  g  being 

positive  integers. 

Then,  §  246,  Prin.  2,  (a"»)«  =  'J^(a«)p 

Case  1,  =  Va^ 

mp 

§  246,  Prin.  1,  =  a  «  . 

Case  3.  —  Let  m  represent  any  value,  and  let  n=—  r^  r  being  a  positive 
integer  or  a  positive  fraction. 

Then,  (a"*)-'-=     ^ 


Casel,  =  — 

§  244,  Prin.  2,  =  «-»»»•. 

Hence,  for  all  values  of  m  and  n,  (a"*)**  =  a»»".  (Ill" 

250.   To  prove  that  the  law  of  exponents  for  evolution  is  general. 

It  is  to  be  proved  that  y/w^  =  a»»^"  for  all  values  of  m  and  n. 

Since  (a'»)'»  =  a*""  for  all  values  of  m  and  n,  it  is  true  when  -  is  sub'- 
stituted  for  n. 

1  1  mxi 

Substituting  -  for  w,  (a™)**  =:  a     ", 

or,  §  246,  >Xa«  ^  a«»^».  (IV) 


246  ACADEMIC  ALGEBRA 

251.  To  prove  (ab)"  =  a"b"  for  all  values  of  n. 

P 
Case  1.  — Let  n  = -,  p  and  q  being  positive  integers. 

Then,  §  249,  Case  1,  since  g  is  a  positive  integer, 

[(a6)?]«  =(«&)?'"' =  (a6)i' 

§  240,  5,  =  oPbP.  (1) 

Also,  §  249,  Case  1,  .  . 

p  p  p  p       p  p    ' 

(jofih^y  =  a^b^  X  a^b^ ...  to  g  factors 

p       p  p       p 

§  83,  =  (a«  X  a?  ...  to  g  factors) (6«  x  6«  ••.  to  g  factors) 

=  aPbP.  (2) 

(1)  and  (2),  Ax.  1,  [(a&)^]^  =  (Jbh^. 

p       p  p 
Taking  the  gth  root,  (^ab)^  =  a^b^. 

Case  2.  —  Let  n  =  -  r,  r  being  a  positive  integer  or  a  positive  fraction. 

Then,  §  244,  Prin.  1,  Cab)-^  =      ^ 

(aby 

Case  1,  =-i— 

§  244,  Prin.  2,  =  a-^b-^. 

Hence,  for  all  values  of  w,    (a&)"  =  a^ft".  (V) 

Examples 

252.  Multiply: 

1.  a^  by  a-\  3.  n*  by  a-\  6.   a^  by  a". 

2.  a^  by  a-\  4.  a  by  a-\  6.    a;^  by  xK 

7.  a^6^  by  a^fei  10.    n-^  by  awt 

8.  7)1^ n  by  m^w~\  11.    a"*-"  by  a""^. 

9.  a*6^  by  a~^b^,  12.    a  ^    by  a  =*  . 

13.  Multiply  x^y~^  -h  o^^  -{-  ^Jr  +  ^V^  +  V^  by  a;^?/^. 

14.  Multiply  ^^  +  a?-y +* -h  a;-y +2  _j_  ^-3^„+3  ^^^  ^„y_„ 


THEORY  OF  EXPONENTS  24? 

15.    Expand  {ah~^  +  1  +  a~h^)  (aV^  -  1  -^  a'h^ 
First  Solution 
ah~^  +      1      +  a'h^ 

ah~^  -    1    +  arh^ 


+  a%^  +  «~^6^  +  arh 
a^6-i  +1  +  a~^6 

Second  Solution 
{ah~^  +  1  +  arhh{ah~^  -  1  +  a"^6^) 
§  97,  =  (a  V^  +  a~h^y  -  l^ 

=  ah-^  +  2  a-^fto  +  arh  -  1 
=  a^6-i  +  2  +  a"^6  -  1 
=  A-i  +  1  +  a"^6. 
Expand : 
16.    (a^+6^)(a^— 6*).         21.    (a;^— a;V^+2/~^(a5*+2/'^)• 
17.    {x^-\-y^{x^-y^).         22.    (a^-6'^+a*6~*+l)(a*-6*). 

18.  (a;-i+10)(a;-Li).       23.    {l-x+x'){x-^+x-^+x-^). 

19.  (a;^-4)(aj^+5).  24.    (a-^+^>"^+c^)(a-'  +  &~^+2c^. 

20.  (a;^_2/t)(a;*+2/^).         25.    (a262_a53_^j4>)(^,-2^-2_|.^-35-i^rt-4^ 

Divide : 

26.  a!^  by  a®.  28.   a*  by  a~l  30.    x^  by  a;^. 

27.  a^  by  a^.  29.    a;^  by  x~^.  31.    a;""^  by  a;"-^. 

32.  Divide  x^  -\-  y^y^  -\-  y^  by  a?'^^. 

33.  Divide  a-^  +  a-%-\-h^  by  a-^ft. 

34.  Divide  a^  +  2  aa^  -i-  3  aV  +  a^a;  —  a*  by  aV. 


248  ACADEMIC  ALGEBRA 

35.   Divide  b-^  +  3  a~^  -  10  a-^b  by  ah-^-2. 

Solution 

aVi  -  2)6-1  +  3  a~^  -  10  «-i6(a"^  +  5  a-^b 
ft-i  _  2  a~^ 

5  a~^  -  10  a-ift 
5  a"^  -  10  a-ift 
Divide : 

36.  a  -  6  by  a^  +  6^  41,    a^  +  2  aV^  +  &"^  by  a*  +  6~i 

37.  a-6bya^-6i  42.    a;^  -  2  +  o:"^  by  aj^  _  ^-f , 

38.  a  +  6bya*  +  &^.  43.    3  -  4.  x-^ -\- x'^  by  x-^  -  8. 

39.  a^H-ft^  by  a*  +  6i  44.    4  aj^^^-i  -  5  2/ -haj-y  by  a^- 2/2 

40.  a;  — 1  by  x^-\-x^-^l.      45.    a^  —  b^  by  a^  4  6^ 

Simplify  the  following : 

46.  (a^y.  49.    (-a^)8.  52.  (8"^)*. 

47.  (a"*)«  50.    (-a')^  53.  (16-^)^. 

48.  (a-y.  51..(-J)-\  54.  (- ^V)"^. 

Expand  by  the  binomial  formula : 

55.  {a^-b^y.      .  57.    (a-'-b^y.  ^59.    (a"^  +  i)^ 

56.  (a^  +  6^)3.  58.    (x~^-yfy.  ^60.    (1  -  a;'/. 

Simplify  the  following : 

61.  V^.  64.  \^a^b-\  67.  (3L  a'^)-^. 

62.  \a~k  65.  V^a^V^-  68.  (i  m-^7r^)l 

63.  \a~i  66.  ^'^6^.  69.  ('ia^y-^^)^. 
Extract  the  square  root  of 

^-  70.    aj2  +  2  a;^  +  3  a;  +  4  a;^  +  3  +  2  a;-^  -f  x-\ 

71.  a;2  +  ^  +  4  z-^  -2xy^-\-A  xz-^  -  4  yV^ 

72.  a  4-  4  6^  +  9  c^  -  4  ah^  -f  6  a^c^  -  12  ^W. 


THEORY  OF  EXPONENTS  249 

Extract  the  cube  root  of 

73.  a2^6a^  +  12a*  +  8. 

74.  a-3aW  +  3a^6^-62. 

75.  8  ic-i  -  12  a;~*2/ -f  6  a;-*2/' -  2/^- 

76.  ic^-6a;H-15a;^-20  +  15a;"^-6a;-i  +  a;~^. 

77.  Factor  4a;~^  — 9^/"^  and  express  the  result  with  positive 
exponents. 

Solution 

§  128,  4a;-2  -  9  2/-2  =  (2  a;-i  +  3  y-^)  (2  a;-i  -  SyO 

'2  .  3\/2 


V«    2//\«    yj 


Factor  the  following  and  express  the  results  with   positive 
exponents : 

78.  a-2-6-2.  83.  J^  -  x-\ 

79.  9-a;-2.  84.  a'^  +  2-\-a-\ 

80.  16 -a-^.  85.  6^-8+16  6-*. 

81.  27-6-3.  86.  12  -  a;-i  -  a;-2. 

82.  h-^-i-y-\  87.  2-3a;-^-2a;-2 

Solve  the  following  equations ; 

88.  x^  =  2.  96.  a;"^  =  6. 

89.  a;^  =  8.  97.  x~^  =  16. 

90.  x^  =  4:.  98.  25a;~^  =  l. 

91.  a;^  =  16.  99.  a;^  =  243. 

92.  ^x^  =  9.  100.  a;^  +  32  =  0. 

93.  x-^  =  5.  101.  aj*  +  a«  =  0. 

94.  \-x^  =  25.  102.  a;^-64  =  0. 

95.  2.T~^  =  JW.  103.  a;"^  +  27  =  0. 


RADICALS 


253.    1.   What  is  indicated  by  V^?  by  a;^^  ^J  Va?  by  ah 

2.  Irdicate  in  two  ways  the  square  root  of  25-,  of  36;  of  2; 
of  3. 

3.  Which  of  these  indicated  roots  can  be  obtained  exactly  ? 
Which  cannot  be  obtained  exactly  ? 

^254.    An  indicated  root  of  an  expression  is  called  a  Radical. 

The  root  may  be  indicated  by  a  radical  sign  or  by  a  fractional 
exponent. 

VSa,  (5a)^,  Va^,  {ax'^y,  Va^  +  2 a&  +  62,  and  (a"^ -\.  2  ab  +  h"^)^  are 
radicals. 

In  the  discussion  and  treatment  of  radicals  only  principal  roots 
will  be  considered. 

'^  255.   The  Degree  of  a  radical  is  indicated  by  the  index  of  the 
root  or  by  the  denominator  of  the  fractional  exponent. 


Va  +  X  and  (&  +  x)^  are  radicals  of  the  second  degree. 

256.  When  the  indicated  root  of  a  rational  number  cannot  be 
exactly  obtained,  the  expression  is  called  a  Surd. 

■V2  is  a  surd,  since  2  is  rational  but  has  no  rational  square  root. 
Vl  +  VS  is  not  a  surd,  because  1  -f  VS  is  not  rational. 

Radicals  may  be  either  rational  or  irrational,  but  surds  are 
always  irrational. 

Both  Vi  and  VS  are  radicals  but  only  VS  is  a  surd. 

257.  An  indicated  even  root  of  a  negative  number  is  called  an 
Imaginary  Number  ;  as  V—  4,  V—  a. 

All  other  numbers,  whether  rational  or  irrational,  are  called 

Beal  Numbers;  as  V25,  V3,  a^,  4. 

250 


RADICALS  251 

258.  A  surd  may  contain  a  i-ational  factor,  that  is,  a  factor 
which  is  a  perfect  power  of  the  same  degree  as  the  radical.  The 
rational  factor  may  be  removed  and  written  as  the  coefficient  of  the 
irrational  factor. 

In  a/8  =  V4  X  2  and  \/54  =  \/27  x  2,  the  rational  factors  are  \/4  and 
V^;  that  is,  V8  =  2V2  and  v/54  =  3v^. 

259.  A  surd  that  has  a  rational  coefficient  is  called  a  Mixed 
Surd. 

2\/2,  a\/^,  and  (a  —  6)  Va  +  6  are  mixed  surds. 

260.  A  surd  that  has  no  rational  coefficient  except  unity  is 
called  an  Entire  Surd. 

\/5,  a/it,  and  Va'-^  +  x^  are  entire  surds. 

261.  A  radical  is  in  its  simplest  form  when  the  expression 
under  the  radical  sign  is  integral,  contains  no  factor  that  is 
a  power  of  the  same  degree  as  the  radical,  and  is  not  itself  a 
perfect  power  whose  exponent  is  a  factor  of  the  index  of  the 
radical. 

y/l  is  in  its  simplest  form  ;  but  \/|  is  not  in  its  simplest  form,  because 
I  is  not  integral  in  form  ;  VS  is  not  in  its  simplest  form,  because  the  square 
root  of  4,  a  factor  of  8,  may  be  extracted  ;  v^25,  or  25^,  is  not  in  its  simplest 
form,  because  25^  =(52)^  =  5^  =  5^,  or  Vl. 

REDUCTION  OP  RADICALS 

262.  To  reduce  a  radical  to  its  simplest  form  when  it  has  a  rational 
factor. 

Examples 


1.    Reduce  V20a®  to  its  simplest  form. 

PROCESS 

V20a'«  =  V4a<^  X  5  =  V4^«  X  V5  =  2a^V5 

Explanation.  —  Since  the  highest  factor  of  20  a^  that  is  a  perfect  square 
is  4a^,  \/20  a^  is  separated  into  two  factors,  a  rational  factor  \/4  a*'\  and  an 
irrational  factor  VS.  V20rr  =  vTo"  x  v^5,  §  220,  Prin.  3.  Extracting  the 
square  root  of  4  a^  and  prefixing  the  root  to  the  irrational  factor  as  a  coefficient, 
the  result  is  2  a^y/Z, 


252  ACADEMIC  ALGEBRA 


2.    Reduce  V  —  864  to  its  simplest  form. 

PROCESS 


V-  864  =V-  216  X  4  =V-  216  X  V4  =  -  6V4 

Rule.  —  Separate  the  radical  into  two  factors  one  of  which  is 
its  highest  rational  factor.  Extract  the  required  root  of  the  rational 
factor,  multiply  the  result  by  the  coefficient,  if  any,  of  the  given 
radical,  and  place  the  product  as  the  coefficient  of  the  irrational 
factor. 

Simplify  the  following : 


3. 

Vl2. 

4. 

V75. 

5. 

-^16. 

6. 

V128. 

7. 

■v/250. 

8. 

^32. 

9. 

V600. 

10. 

V500. 

11. 

-^160. 

12. 

-v^SOOO. 

13. 

-^81. 

14. 

^/189. 

15.  V162.  X27.    V243aV». 

16.  Vl8^.  28.    -v/128  a%\ 


17.    V25&.  29.    V405ay. 


18.  V98^._  30.  V375a5y. 

19.  V50a.  31.  (245 aV')^- 

20.  ^640.  32.  (135a;y)l 

21.  V84.  33.  {a^  +  ^a?)^. 

22.  \/72.  34.  (16  a; -16)^ 

23.  ^192. 

24.  V800. 


35. 

Vl8a 

;-a 

36. 

^^- 

-2a5«. 

37. 

V8- 

20  h\ 

25.  V3645. 

26.  V735.  38.    5 V4 a^  +  4. 

39.  V5a:2-10iC2/  +  52/^.  41.    (3am2+ 6  am +  3a)i 

40.  V4a3-24a2a;_|_36aa^.      42.    {x^y  ~  3  s?y'^  +  ^  x^f  -  xy^)^ . 

43.    Reduce  -v/^r-^  to  its  simplest  form. 

PROCESS 


RADICALS 


253 


Explanation.  —  Since  a  radical  is  not  in  its  simplest  form  when  the  ex- 
pression under  the  radical  sign  is  fractional,  the  denominator  is  to  be  removed ; 
and  since  the  radical  is  of  the  second  degree,  the  denominator  must  be  made 
a  perfect  square.  The  smallest  factor  that  will  accomplish  this  is  2  y.  Multi- 
plying the  terms  of  the  fraction  by  this  factor,  the  largest  rational  factor 

of  the  resulting  radical  is  found  to  be  \/-^,  which  is  equal  to  -^-    There- 

\  4  »/4  2  1/2 


f4y4 

fore,  the  irrational  factor  is  V2y,  and  its  coefiBcient  is  -^• 
Simplify  the  following : 


2y2 


52. 


45.  Vi 

46.  Vi 

47.  Vi. 

48.  V|, 

49.  V| 

50.  </^. 

51.  n 

eo.   (.^^)^g|. 

2?/        jx-2y 
\     2w 


2a' 


56. 


\E 


53.  ^J^. 

M  2  0" 


57. 


4a 

3iB2" 


54 


55 


■4 


58 


rsx_ 

\50a^ 


'y 


69. 


61 


a:-22/^     22/ 


62.    (l-a^)^/I^i±^. 

(g  +  bf  3J  a-{-b 
a-b    \(a-by' 


63. 


263.  To  reduce  a  radical  to  its  simplest  form  when  the  expression 
under  the  radical  sign  is  a  perfect  power  of  a  degree  corresponding  to 
some  factor  of  the  index  of  the  root. 

Examples 

1.  Reduce  VOci^  to  its  simplest  form. 

PROCESS 

2.  Keduce  V64a''6^^  to  its  simplest  form. 

PROCESS 

-5^64  aV  =  >/2VW  =  6  (2  ab)^  =  6  (2  ab)^  =  b  -s/IoFb^ 


254  ACADJEMIC  ALGEBRA 

Simplify  the  following : 

3.  V^.  7.  vTeOO.  11.  W^W^. 

4.  a/25.  8.  ^/27^.  12.  a/121  aV. 

5.  A/ii4.  9.  A^MS.  13.  Va'h^c'd\ 

6.  ^81.  10.  ^289.  14.  ^{x'-'Ixy+f). 

264.    To  reduce  a  mixed  surd  to  an  entire  surd. 
Examples 

1.  Express  2  a  V5  6  as  an  entire  surd. 

PROCESS 

2aV5l)  =  \/4^V56=V4a2x56=V20^ 

Rule.  —  Raise  the  coefficient  to  a  power  coiTesjJonding  to  the 
index  of  the  given  radical,  and  introduce  the  result  under  the  radical 
sign  as  a  factor. 

Express  as  entire  surds : 

2.  2V2.  6.    3^/3.  10.  iV2.  14.  fVif. 

3.  3V5.  7.    4V5.  11.  fV^.  15.  fVff^. 

4.  5V2.  8.    -^VS.  12.  ^VbE.  16.  i^/I^. 

5.  3-v/2.  '  9.    a^-^.  13.  fV|.  17.  Iv'Sf. 


18.    ^^±IJ^^.        19.    ^JlZUZ.        20.    l(a-5)l 
a;_2/\a;  +  2/  a  —  4>'        a  +  4  a6^  ^ 

265.  To  reduce  radicals  of  different  degrees  to  equivalent  radicals 
of  the  same  degree. 

1.  Express  a^  by  an  equivalent  radical  with  an  exponent  in 
higher  terms. 

2.  What  is  the  degree  of  the  radical  x^?  Express  x^  as  a 
radical  of  the  12th  degree.  Express  x^  as  a  radical  of  the  12th 
degree.     Express  6^  and  b^  as  radicals  of  the  same  degree. 

Examples 

1.  Eeduce  V3,  V2,  and  ^i  to  equivalent  radicals  of  the 
same  degree. 


RADICALS  265 

PROCESS 

</3  =  3i  =  3A=^'3^  =  ^27 

^  =  4^  =  4^  =  '^4^  =  ^256 

Rule. — Express  the  given  radicals  with  fractional  eoeponents 
having  a  common  denominator. 

Raise  each  number  to  the  power  indicated  by  the  7iumerator  of  its 
fractional  exponent,  and  iiidicate  the  root  expressed  by  the  common 
denominator. 

Reduce  to  equivalent  radicals  of  the  same  degree : 

2.  V2  and  </3.  9.  V^,  Va^,  and  ■\/2. 

3.  V5  and  V6.  10.  Va,  Vft,  V^,  and  Vj/. 

4.  -Vl  and  VIO.  11.  ^a  +  6  and  Vaj-hy. 

5.  a/IO,  V2,  and  ^5.  12.  V|,  \/S^,  and  2V5. 

6.  Vi,  ^2,  and  V3.  13.  -y/x,  y/xy,  and  Va?y\ 

7.  ^13,  V5,  and  ^4.  14.  (a  +  fe)Va^^  and  ^/aTIT^. 

8.  V3,  ^,  and  \/^.  15.  Va+&, -v/^+P^  and  Va^. 

ADDITION  AND  SUBTRACTION  OF  RADICALS 

266.  Radical  terms  that,  in  their  simplest  forms,  are  of  the 
same  degree  and  have  the  same  number  under  the  radical  sign 
are  called  Similar  Radicals. 

267.  Principle.  —  Only  similar  radicals  can  be  united  into  one 
term  by  addition  or  subtraction. 

Examples 

1.   Find  the  sum  of  VSO,  2-5^8,  and  6V|. 

T>  x> /~v  ri  xj*  c  o 

Explanation.  —  To  ascertain  whether  or  not  the  given 
-y/50  =:    5  -y/2     expressions  are  similar  radicals,  each  may  be  reduced  to 

rt  «/^ 9/9     ^^  simplest  form.     Since,  in  their  simplest  form,  they  are 

of  the  same  degree  and  have  the  same  number  under  the 
6  V Y  =  3  V  2  radical  sign,  they  are  similar,  and  their  sum  is  tlie  sum 
"^        _'if\    /rt     of  the  coefficients  prefixed  to  the  common  radical  factor. 


256  ACADEMIC   ALGEBRA 

Find  the  sum  of 

2.  A^50,  Vl8,  and  V98. 

3.  V27,  Vl2,  and  V75. 

4.  V20,  V80,  and  V45. 

5.  V28,  V63,  and  VTOO. 

10.  Vl,  VI25,  VJ,  and  Vii- 

11.  VJ,  V75,  |V3,  and  Vl2. 

12.  v1,  iV3,  |-v^9,  and  Vl47. 

13.  -JOO,  V28,  -v/25,  and  a/TTS. 

14.  ^375,  V44,  ^192,  and  V99. 
Simplify : 

"15.  V245-Vi05+V45. 

16.  v'12 -f- 3V75  -  2 V27. 

17.  5V72  +  3Vi8-V50. 
_18.  ■^128  4--v/686-v^. 

19.  Vll2-V3l3H-V448. 

20.  ^135-^625+^320. 

21.  ^f  +  ^4-^. 

22.  ■v/864-^4000  4--</32.// 

28.    6^p  +  4^-8^|||, 


6.  -v/250,  ^16,  and  -y/M. 

7.  -v/IM,  a/686,  and  ^/J. 
^I35,  -^320,  and  -\/625. 


8. 


9.    -v/500,  ^^108,and-v/-32. 


23.    -v^28^4-^/375^-A/54ic. 


2/'       ^z 
Uax" 


4  ga;^ 
6/ 


27.    V(a+6)2c-V(a-6)2c. 
29.    ■</-  96  x'  +  2^/3^  -  a/5^  +-v/40^. 


30.    Vafta;  -  Va^feV  +  V8  a^6V. 


31     V3^T30^TT5^-V3a^-6aj2-f  3  a;. 


3g.    V5a*  +  30a*  +  45a3-V5a^-40a^  +  80a3. 

33.  V50+-^9-4VJ  +  -v^^=^24+-^^-a/64. 

34.  V|-h6V|-iVl8+-v/36--v/J|  +  A/T25-2V^. 


RADICALS  257 

MULTIPLICATION   OF   RADICALS 

I.    1.    What  is  the  exponent  of  a  in  a^  x  a^?  in  a^  x  a*? 
in  a^  X  a^?    in  a^  x  a^?   in  a^  x  a^  ?  in  Va  X  ^  ? 

2.  When  the  fractional  exponents  indicate  different  roots,  what 
must  be  done  before  the  radicals  can  be  multiplied  together  ? 

Examples 
PROCESSES.  —  1 .    VT  X  V5  =  V35 

2.  5V3x2Vi5  =  10V45  =  10x3V5  =  30V5 

3.  2V3  X  3^  =  2</27  X  3^  =  6 v'iOS 

Rule.  —  If  necessary,  reduce  the  radicals  to  the  same  degree. 

Multiply  the  coefficients  together  for  the  coefficient  of  the  product 
and  the  factors  under  the  radical  sign  for  the  radical  factor  of  the 
product,  and  simplify  the  result. 

Multiply : 

4.  V2  by  Vs.  13.  2V6  by  VlS.' 

5.  V2  by  V6.  14.  2^3  by  3^45. 

6.  V3  by  Vl5.  15.  2^6  by  3V6. 

7.  V3  by  V48.  16.  3V3  by  2^5. 

8.  2V5  by  3V10.  17.  ^5  by  VlO. 
^^9.  3V20  by  2V2.  18.  2^250  by  V2. 

10.  V2  by  3-v/3.  19.    2^24  by  Vl8. 

11.  V2  by  2V5.  20.    V28  by  3V7. 

12.  V3  by  3V3.  21.   2^2  by  V512. 

Find  the  value  of 

22.  Voft  X  Vftc  X  Vcd  X  Vda. 

23.  V^  X  Vl2^  X  V75  xy\ 

24.  V2^  X  Va6c  X  V4a262. 

ACAD,    ALG.  — 17 


258 


ACADEMIC  ALGEBRA 


25.  Vm7i  X  s/m^n  x  Vmn^ 

26.  V2  axy  x  \/^  X  A/o?xy. 

27.  Vx-^y  X  "v^cc-y  X  Va;~y. 

28.  Va  -  5  X  a/o^^^  X  \/(a  -  &)-2. 

29.  V|xVJxV|.  32.    -V^i  X -5/f  X  V|. 

30.  VJxV|xV|.  33.  a/|xa/|xV|. 

31.  -v/lx-x/fxVI  34,     ^|XV|X^|. 
35 .    Multiply  2  V2  +  3  V3  by  5  V2  -  2  V3. 

Solution 
2\/2  +  3\/3 
6V'2-2V3 


20  +  isVe 

-    4V6-18 


Multiply : 


20  +  ll\/6- 
=    2  +  ll\/6. 


18 


36.  V5  4-V3  by  V5-V3. 

37.  V7  +  V2  by  V7  -  V2. 

38.  V6-V5  by  V6-V5. 

39.  5--V5  by  l+VS. 

40.  4V7  +  1  by  4V7-1. 

41.  2V2+V3  by  4V2+V3. 

42.  2 V3  +3V5  by  3 V3  +  2 V5, 

43.  3a+V5  by  2a-V5. 

44.  2V6-3V5  by  4V3-VI0. 

45.  a^-  a6V2  +  b^  by  a^  +  a6V2  +  b\ 

46.  0"  —  Vxyz  +  ?/2  by  Va;  +  Vyz. 

47.  aj Va;  —  x^y  +  ?/ V^  —  2/ Vy  by  Vx  +  Vy. 


RADICALS  259 

DIVISION  OP  RADICALS 

269.    1.  What  is  the  exponent  of  a  in  a^-^a^?   in  a*-7-a*? 

in  a^^ah   in  af'^  ^  ah   in  Va^^? 

2.    When  the  fractional  exponents  indicate  different  roots,  what 
must  be  done  before  one  radical  can  be  divided  by  another  ? 

Examples 
PROCESSES.  —  1 .    V60  -7-  Vl2  =  V5 

2.    ^2-V2=^4--^8=^|  =  ^  =  ^^/32 

Rule.  —  If  necessary ,  reduce  the  radicals  to  the  same  degree. 
To  the  quotient  of  the  coefficients  annex  the  quotient  of  the  radical 

factors  under  the  common  radical  sign,  and  reduce  the  result  to  its 

simplest  form. 


Find  the  quotient  of 

4.    V50-V8. 

12. 

2^J^--V8. 

5.    V72-2V6. 

13. 

Vax  -T-  Vxy, 

6.   4V5-V40. 

14. 

V2a¥~r--y/a'b\ 

7.    6V7-V126. 

15. 

•■v/aV-V2aa;. 

•^8.    -V/4-V2. 

"^16. 

^/9a'b'^VSab. 

9.    7\/135V^/9. 

17. 

■\/4:afy'-^</2xy. 

10.    7V75H-5V28. 

18. 

Va  —  6  -f-  Va  +  b. 

11     -J^H-v/32. 

19. 

3v^-V|. 

20.  Divide  V15-V3  by  V3. 

21.  Divide  V6-2V3  +  4  bjr  V2. 

22.  Divide  V2  +  2+iV42  by  ^V6. 

23.  Divide  5V2  +  5V3  by  Vl0+Vl5. 

24.  Divide  5  +  5 V30  +  36  by  V5  +  2 V6. 


260 


ACADEMIC  ALGEBRA 


INVOLUTION  AND  EVOLUTION  OP  RADICALS 

270.   In  finding  powers  and  roots  of  radicals  it  is  frequently 
convenient  to  use  fractional  exponents. 

Examples 

1.  What  is  the  cube  of  2Va^? 

Solution.      (2  Vax3)8  =  2^(ax^)^  =  8  aW  =  8  VcS  =  8  ax^Vax.  ) 

2.  What  is  the  square  of  3^v^  ? 

Solution.      (3\/x5)2  =  9(a^)^  =  9x^  =  9v^^  =  9  a;  v^. 

3.  What  is  the  cube  of  V2  +  1  ? 

Solution 

(V2  +  1)3  =(V2)8  +  3(\/2)2  .  1  +  3V2  .12+18 

=  2V2  +  6  +  3V2+l 
=  7  +  5  V2.  : 

In  such  cases  expand  by  the  binomial  formula. 


Square : 

4.  3Va6. 

5.  2^/3^. 

6.  x^j2^. 

7.  nV4&. 

8.  a<fo?b. 


Cube: 

9.    2V5. 

10.  3V2. 

11.  2^^. 

12.  -J/o^. 

13.  -S/i^. 


Involve  as  indicated: 

14.  (-2V2a6)\ 

15.  (-V2v/x)l 

16.  (-V2v/^)*. 

17.  {-2V^^y)\ 

M    n 

18.  (-3aV)6^ 


Expand : 

19.  (2+V6)2. 

20.  (2+V2)2. 

21.  (2+^/5)^. 


22.  (2-V3)^ 

23.  (V7-^/6)2. 

24.  (2V2-V3)2. 
28.    What  is  the  cube  root  of  —  27  Va^? 
Solution.          ^ -  27Vax={-  21)^(axy  =-SVcac. 


25.  (V^±l)*. 

26.  (Va-^/b)\ 

27.  V^±l)^ 


RADICALS  261 

29.  What  is  the  fourth  root  of  V2^  ? 
Solution.  a/  V2x  =  [(2 x)^]^  =  (2 x)^  =  y/2x. 

Find  the  square  root  of  Find  the  cube  root  of 

30.  V2.         33.    -V^.  36.    ■\/2x.  39.    -21^'a^. 

31.  ^5.         34.     v^.  37.    Vfa^  40.    -  64a/^. 

32.  -v^.        35.    VaV.  38.    "^8  m^ar*.       41.    -Vo^. 

Simplify  the  following  indicated  roots : 

42.  V-^4^^.  44.    (VSo^^*.  // 2^^2X2 

1  46.    -\/( — : — )«• 

43.  a/Vo^.  45.    (V^^)"^.  ^\<^~'YJ 


RATIONALIZATION 

1  V^ 

271.   How  may  the  fraction be  reduced  to  the  form  -^— ? 

■^  _       V2  2 

Ato^?    -2-to?^?    ^to^? 

V2  2  V5  5  V^  ^ 

^^  272.   The  process  of  rendering  a  surd  expression  rational  is 
called  Rationalization. 

The  factor  by  which  a  surd  expression  is  multiplied  to  render 
it  rational  is  called  the  Rationalizing  Factor. 

The  denominator  of  — ^  is  rendered  rational  by  multiplying  it  by  VS. 

V3 

Thus,  —  = .     \/3  is  a  rationalizing  factor  for  the  denominator. 

V3        3 

273.  A  binomial,  one  or  both  of  whose  terms  are  surds,  is  called 
a  Binomial  Surd. 

2  +  \/3,  Vx  +  Vy,  and  a*  —  6*  are  binomial  surds. 

274.  Two  binomial  surds  of  the  second  degree  whose  product 
is  rational  are  called  Conjugate  Surds. 

Va  +  V&  and  Va  —  Vft  are  conjugate  surds ;  also  a  +  y/h  and  a  —  y/h. 
Conjugate  surds  differ  only  in  the  sign  of  one  of  the  terms. 


262  RADICALS 

y^l  275.   Principle.  — A  binomial  surd  of  the  second  degree  may  be 

rationalized  by  multiplying  it  by  its  conjugate. 

Thus,  the  product  of  a  +  Vb  and  a  —y/b  is  a^  —  b. 

Examples 

1.  Find  the  simplest  rationalizing  factor  for  V3  x. 
Solution.  VSx  x  V3  a;  =  VO  a:-^  =  3  x. 

.'.  VSx  is  the  simplest  rationalizing  factor. 

2.  Find  the  simplest  rationalizing  factor  for  V4  a. 
Solution.  \/4  a  x  V2  a'^  =  VS  a^  =  2  a. 

.'.  \/2  a^  is  the  simplest  rationalizing  factor. 

Find  the  simplest  rationalizing  factor  for 

3.  V3.  5.    Va^.  7.    -VWx.  9.    ^^4. 

4.  V6.  6.    V4a.  8.    V8.  10.    -^a^, 
11.   Find  the  simplest  rationalizing  factor  for  V5  +  V3. 
Solution.     (  V5  +  V3)  (  Vs  -  V3)  =  (  y/6y -  (  V3)2  =  5  -  3  =  2. 

.'.  V5  —  V3  is  the  simplest  rationalizing  factor. 

Explanation. — Since  ( V5)2  =  5  and  (^3)2  =  3,  V5  +  \/3  may  be 
rationalized  by  multiplying  it  by  some  factor  that  will  give  the  square  of 
each  term,  but  no  other  terms.  Since  the  product  of  the  sum  and  difference 
of  two  numbers  is  equal  to  the  difference  of 'their  squares,  the  simplest 
rationalizing  factor  for  V5  +  VS  is  its  conjugate,  V5  —  a/3  (Prin.). 

/  12.    Find  the  simplest  rationalizing  factor  for  Va  4- V&^. 

/  Solution.  — By  §  111,  Va  +  \/p,  or  a^  +  &t,  is  exactly  contained  in  the 
sum  of  any  like  odd  powers  of  a^  and  ftt,  and  also  in  the  difference  of  any 
like  even  powers  of  a^  and  fet.  Since  in  raising  a^  and  b^  to  the  same  power 
the  exponents  |  and  §  are  multiplied  by  the  index  of  the  power,  the  lowest 
like  powers  of  ai  and  b^  that  are  rational  numbers  are  the  sixth  powers, 
which  are  even  powers.  Hence,  the  rational  expression  of  lowest  degree  in 
which  a^  +  b^  is  exactly  contained  is  (a^)^  -  (6^)^,  or  a^  —  5*. 

Dividing  a^  —  ¥  by  ai  +  &^,  or  by  Va  +  y/b'^,  the  rationalizing  factor  is 
found  to  be  at  -  a^&f  +  ah^  -  ab^  +  ah^  -  b^. 


RADICALS  263 

Find  the  simplest  rationalizing  factor  for 

13.  V3-fV2.  16.    Va-hV«.        •      19.  / -v^o^  + -v/fe. 

14.  V3  -  V2.  17.    a-2  VS.  20.  \  Va  -  Vx. 

15.  2  +  V3.  18.    3Vx  +  2?/.  21.    V^+-\^. 

3  V 

22.   Rationalize  the  denominator  of 

Vl2 
Solution 
3    _     3  X  V3     ^3V3_3y3_\/3 
.  Vl2  ~  Vl2  X  V3      V36         6  2  ' 


or  ^V3. 


H-ationalize  the  denominators : 

23.    ±,.  26.    ^.  29.    -4-  32.      ^ 


V3  V6  V*'  ^cKC^ 


24.    A.  27.    -^.  30.    -^.  33.    ^' « +  & 


V5  -v/4  V2ax  Va-6 

4 


25.    ^/-.  28.    ?:^.  31.    -^.  34.    ^. 

V6y  a/12  -v^l^ 

V7-V3 


35.    Rationalize  the  denominator  of 

V7  + V3 

Solution 

/7_V3^(V7_V3)(V7-\/3)_7-2V^+3_10-2v^_5-V^ 
v/7+V3~(V7  4-V3)(V7-V3)  7-3  4  2 

Rationalize  the  denominators : 

X  36.    ^^?+^.  38.       ^-^.  40.    AZ^^. 

V3  -  V2  V2  +  V3  V8  -  V7 

37.    5_.  39.    ^--^A  41.    V^+V^. 

V5  -  V3  2  -  V2  ^x-Vy 

42.    4V2  +  6V3  44^    a;-V^2TrT^ 

3V3-2V2   X'  *   x-^-Vx--l 


A  o     Va;  +  1  —  2  .  _     aVa  —  Va;  +  1 

43.     — zmiz= •  45. ■ 

Va;  +  l  +  2  Va3  +  Va;  +  1 


264  RADICALS 


4e.  Va?  +  y-Va:-y.  47^  Vo^^  +  g  +  i-l, 

V«4-y  4-  Vit "  2/  Va^  H-  a  +  1  +1 

48.  Find  the  approximate  value  of 

Solution 
_^^5V3^5x  1.73205^, 
V3        3  3 

In  solving  the  following  examples,  the  work  will  be  lessened  by  observing 
that  V2  =  1.41421,   V3  =  1.73205,  and  V6  =  2.23607. 

Find  the  approximate  value  of 

49.  -^.  52.    ^.  55.    8-^ 


V2  V45  2-V3 

60.    A.  63.    -1^.  56.    1+^. 

V6  V50  2-V2 

51.    ±..  54.    -4..  57.    ^±2^^. 

V8  V125  5  -  2  V5 

58.    Simplify  ^^-^^-^~^, 
V2+V3+V5 

Solution 

Since  (  V2  4- V3)  -  VS  =  (  \/2  -  \/5)  +  \/3, 

V2-V3-V5^(\/2-V5)-V3^^  (\/2-V5)  +  \/3 
V^  +  V3+\/5      (V2+V3)  +  \/5      (V2+V3)-\/5 
_2-2\/l0  +  5-3^4-2Vl0 

2  +  2V6  +  3-5  2V6 

^2->/l0^2v^-2\/T5^  v^-\/l5 
V6  6  3         ' 

Rationalize  the  denominators : 

69.    ^-^-^^.  l4l. 


V2H-V5+V7  V2+V3+V5 

60  V3+_V2  y^g     2V2-3V3  +  4V5 

V34-V2-V6  '       V2+V3-V5 


RADICALS  265 

276.  To  find  the  square  root  of  a  binomial  surd. 

( V2  +  V6)2  =  2  H-  2  Vl2  4-6=8  +  2  Vl2 ; 
( Vi  +  V3)2  =  4  +  2  Vl2  +  3  =  7  +  2  Vl2 ; 
or  (2+V3)2  =  4  +  4V3    +3  =  7  +  4V3. 

1.  Since  (V2  +  V6)^  =  8 +2  Vl2,  what  is  the  square  root  of 

8  +  2V12? 

2.  How  does  the  product  of  the  terms  of  the  square  root  of 
8  +  2  Vl2  compare  with  the  irrational  term  2  Vl2  ? 

3.  How  does  the  sum  of  the  squares  of  the  terms  of  the  square 
root  compare  with  the  rational  term  8  ? 

4.  How,  then,  may  the  square  root  of  8  +  2  Vl2  be  found  from 
the  termsof  8  +  2Vi2? 

5.  How  may  the  terms  of  the  square  root  of  7  +  2  Vl2,  or  the 
equivalent  expression  7+4  V3,  be  found  ?  After  the  irrational 
term  is  divided  by  2,  what  two  factors  of  the  result  are  selected 
for  the  terms  of  the  root  ? 

277.  A  surd  of  the  second  degree  is  called  a  Quadratic  Surd. 

Vi,  4  ViC,   Vx  +  Vy,  and  3  -f  2  \/5  are  quadratic  surds. 

278.  Principle.  —  The  terms  of  the  square  root  of  a  quadratic 
binomial  surd  that  is  a  perfect  square  may  be  obtained  by  dividing 
the  irrational  term  by  2  and  then  separating  the  quotient  into  two 
factors,  the  sum  of  whose  squares  is  the  rational  term. 

Examples 
1.   Find  the  square  root  of  14  +  8  V3. 

Solution 

Since,  if  14  -f-  8  V3  is  the  square  of  a  binomial  quadratic  surd,  the  irra- 
tional term  8  V3  is  ttcice  the  product  of  the  terms  of  the  root  (Prin.),  4  VS, 
or  •v/48,  is  the  product  of  the  terms  of  the  binomial  surd.  Since  the  two 
factors  of  \/48,  the  sum  of  whose  squares  is  14,  are  Vd  and  Vs,  the  required 
square  root  is  equal  to  \/6  +  V8. 


.V  Vl4  +  8  V3  =  \/6  +  V8. 


266  ACADEMIC  ALGEBRA 

2.   Find  the  square  root  of  11  —  6  V2. 

Solution 


Vll-6V2=Vll  -2Vl8. 
=  3-V2. 

Find  the  square  root  of  each  of  the  following : 

3.  12  +  2V35.  11.  12 +  4  Vs. 

4.  16-2V60.  12.  11  +  4V7. 

5.  15  + 2  V26.  13.  12-6V3. 

6.  16- 2  V55.  14.  17  +  12V2. 

7.  11+2V30.  15.  15-6V6. 

8.  7-2ViO.  16.  18H-6V5. 

9.  3-2V2.  17.  a2  +  6  +  2aV6. 
10.  6  +  2V5.  18.  2a-2-Va'-b^ 

PROPERTIES  OP  QUADRATIC  SURDS 

279.    The  square  root  of  a  rational  number  cannot   he  partly 
ratioyial  and  partly  a  quadratic  surd. 

For,  if  possible,  let  Vy  =  V5  ±  m. 

By  squaring,  y  =  6  ±  2  mVft -f  m^, 

and  ^l^^y-^n^-b. 

2  m 

that  is,  a  surd  is  equal  to  a  rational  number,  which  is  impossible.    Therefore, 
vV  cannot  be  equal  to  Vb  ±m. 

'  280.  In  any  equation  containing  rational  numbers  and  quadratic 
surds,  as  a  -\-  Vb  =  x  -{- '  Vy,  the  rational  parts  are  equal,  and  also 
the  irrational  parts. 

Let                                       a-\-Vh  =  x-\-Vy.  (1) 
Since  a  and  x  are  both  rational,  if  possible,  let 

a  =  x±m.  (2) 

Then,                         x  ±  m  ■{■  s/h  =  x -{-  Vy,  (8) 

and                                                Vy  =  V&  ±  w».  (4) 


RADICALS  267 

Since,  §  279,  equation  (4)  is  impossible,  a  =  x±m  is  impossible ;  that  is, 
a  is  neither  greater  nor  less  than  x. 

Therefore,  a  =  x^   and,   Eq.  (1),  Vh  =y/'y. 

Hence,  if  a  -^-Vb  =  x-{-  Vy,  a  =  x,  and  y/h  =  Vy. 


^281.   If  yla-\--Vb  =  V^  +  Vy,  then  \'a  —  V6  =  Va;  —  Vy,  when 
a,  b,  X,  and  y  are  rational  and  a  >  Vb. 

For,  squaring,  a  +  Vft  =  x  +  2  Vxy  +  y. 

Therefore,  §  280,  a  =  x  +  y,  and  Vb  =  2  Vxy. 

Hence,  a  —  Vb  =  x-\-y  —  2  Vxy. 

Whence,  V  a  —  Vb  =  Vx  —  y/y. 

Examples 
1.    Find  the  square  root  of  21  +  6  VlO. 

Solution 


Let  .    Vx  +  Vy  =  V2I  +  6 VlO.  (1) 


Then,  §  281,  Vi  -  Vy  =  V2I-6VTO.  (2) 


Multiplying,  x-y=  V441  -  360  =  V81, 

or  x-y  =  9.  (3) 

Squaring  (1),  x -\-2y/xy  +  y  =  21  ■\- Qy/TO. 

Therefore,  §  280  x-\-y  =  2l.  (4) 

Solving  (4)  and  (3),  x=  15,  y  =  6. 

.-.  Vx=VT6,   Vy=V6. 
Hence,  the  square  root  of  21  +  6  VTO  is  Vl5  +  V6. 

Find  the  square  root  of 

2.  25  +  IOV6.  8.  16-f-6V7.  14.  2+V3. 

3.  19-h6V2.  9.  21-8V5.  15.  6+V35. 

4.  454-3OV2.  10.  47-I2VTT.  16.  I  +  IV2. 

5.  35-14V6.  11.  56  4-32V3.  17.  2  4- f  V6. 

6.  ll-}-6V2.  12.  35-12V6.  18.  30  +  20 V2. 

7.  24-8V5.  13.  5G -12v'3.  19.  18 -G\/5. 


268  ACADEMIC  ALGEBRA 

RADICAL  EQUATIONS 

282.  An  equation  involving  an  irrational  root  of  an  unknown 
number  is  called  an  Irrational,  or  Radical  Equation. 

x^  =  3,   Vx  4-  1  =  Vx  —  4  +  1,  and  y/x  —  1  =  2  are  radical  equations. 

283.  An  equation  containing  quadratic  surds  involving  x  may 
be  rationalized  with  respect  to  x  by  writing  all  the  terras  in  the 
first  member  and  multiplying  both  members  by  the  proper  ration- 
alizing factor. 

To  rationalize  Vx  —  3  =  2.  (1) 

Transposing,  Vx  —  3  —  2=0.  (2) 

Multiplying  by  the  conjugate  surd, 

(Vx^^-2)(Vx^^  +  2)=0,  (3) 

or  X  -  3  -  4  =  0.  (4) 

The  rationalization  of  such  equations,  however,  is  more  con- 
veniently accomplished  by  the  process  of  squaring. 

Thus,  by  squaring  (1),  (4)  may  be  obtained  in  the  form 

x-3  =  4.  (5) 

It  is  evident,  then,  that  squaring  an  equation  is  equivalent  to 
multiplying  both  members  by  the  same  unknown  expression,  an 
operation  likely  to  introduce  roots  (§  196).  The  roots  introduced, 
if  any,  when  an  equation  is  rationalized  are  those  of  the  equation 
or  equations  formed  by  placing  each  rationalizing  factor  equal  to 
zero. 


Thus,  in  squaring  the  equation  vx  —  3  =  2,  the  root  of  Vx  —  3  =  —  2  is 
introduced.  But  if  \/4  is  taken  to  mean  either  +  2  or  —  2,  this  equation  has 
the  same  root  as  the  given  equation  and  no  root  has  been  introduced. 

Repeated  squaring  is  often  necessary  to  free  an  equation  of 
quadratic  surds  involving  x.  This  corresponds  to  repeated  ration- 
alization with  respect  to  particular  surds. 

Thus,  Vx  _  5  =  _  Vx  +  5. 

Squaring,  x  -  10  Vx  +  25  =  +  (x  +  5)  =  x  -f-  5. 

Simplifying,  \^  =  2. 

Squaring,  x  =  4. 

Or,  transposing  in  the  given  equation, 

Vx  -  5  +  Vx  +  5  =  0. 


RADICALS  269 

Rationalizing  with  respect  to  Vx  +  5, 

(Vx-  5  +  Vx  +  b){Vx-  5  -  Vx  +  5)  =  0. 

Expanding,  x  —  10  Vx  +  25  —  (x  +  5)  =  0. 

Simplifying,  Vx  -  2  =  0. 

Rationalizing,  (  Vx  —  2)  (  Vx  +  2)  =  0, 

or  X  -  4  =  0. 

In  squaring  the  first  time,  the  factor  Vx— 5— Vx+5  is  introduced;  and 
in  squaring  the  second  time  the  factor  Vx+2,  which  is  ^j^  of  the  product 
(\/x+5+  \/x+5)(Vx  +  5  -  Vx  +  5),  or  10  Vx  +  20,  is  introduced. 

284.    It  follows  from  thie  preceding  discussion  that : 
If  a  radical  equation  is  rationalized  by  multiplying  by  a  rational- 
izing factor  or  by  squaring,  the  resulting  equation  has  all  the  roots 
of  the  given  equation. 

Whether  the  given  radical  equation  has  all  the  roots  of  the 
rational  equation  depends  upon  the  method  agreed  upon  of 
verifying  radical  equations. 

As  illustrated  above,  each  of  the  equations 

Vx  -  5  +  Vx  +  5  =  0,  (1) 

Vx  -  5  -  Vx  +  5  =  0,  (2) 

Vx  +  5  +  Vx+5  =  0,  (;j) 

and  Vx  +  5  -  Vx  +  5  =  0  (4) 

is  rationalized  by  finding  the  product  of  them  all,  which  is  x— 4=0.  Hence, 
the  equation  x  —  4  =  0  has  the  roots  of  the  four  equations ;  that  is,  each 
equation  has  the  root  x  =  4,  or  has  no  root. 

If  X  =  4,  Vx=V4:  and  Vx  +  5  =  Vo.  If  Vi  is  either  +  2  or  -  2  and  V9 
is  +  3  or  —  3,  the  equations  are  verified  as  follows  : 

(1)  becomes  2-5  +  3  =  0, 

(2)  becomes  2-5-(-3)=0, 

(3)  becomes  — 2  +  5+(-3)=0, 

(4)  becomes  _2  +  5-3  =  0. 

To  prevent  confusion  in  making  numerical  substitutions,  it  is 
customary  to  regard  only  the  positive  or  principal  square  root  in 
expressions  like  V4,  V9,  VS,  etc.     (See  §  225.) 

For  example,  by  common  agreement  +  V9  means  +(+  3),  or  +  3,  —  V9 
means  —(+3),  or  —3;  +V5  means  the  positive  square  root  of  5;  etc. 
With  this  understanding  equations  (2),  (3),  and  (4)  cannot  be  verified  for 
X  =  4,  and  since  they  have  no  other  root,  they  may  in  this  sense  be  regarded 
as  impossible  equations. 


270  ACADEMIC  ALGEBRA 

Wlien  the  equations  given  in  this  section  have  been  freed  from 
the  radical  signs,  the  resulting  equations  will  be  found  to  be 
simple  equations.  Other  varieties  of  radical  equations  are  treated 
subsequently. 

Examples 

1.   Given  V2  a;  +  4  =  10,  to  find  the  value  of  x. 

Solution 

Transposing,  V2x  =  6. 

Squaring,  2  aj  =  36. 

.-.  x  =  18. 


2.    Given  s/x  —  7  +  V»  =  7,  to  find  the  value  of  x. 

Solution 
Vx-  7  +  Vx  =  7. 
Transposing,  Vx  —1  =  7—  Vx. 

Squaring,  x  —  7  =  49  —  14\/x  4-  as. 

Transposing  and  combining,     14  Vx  =  56. 
Dividing  by  14,  Vx  =  4. 

Squaring,  x  =  16. 


3.   Given  \  14  +  Vl  +  Va;  +  8  =  4,  to  find  the  value  of  x. 

Solution 


\14+Vl+VxT8  =  4. 


Squaring,  14  +  Vl  +  Vx  +  8  =  16. 


Transposing,  etc.,         Vl  +  Vx  +  8  =  16  -  14  =  2. 
Squaring,  1  +  Vx  +  8  =  4. 


Transposing,  etc.,  Vx  +  8  =  4  —  1  =  3. 

Squaring,  a;  +  8  =  9. 

.-.  x  =  9-8  =  l. 

Vebification.  \l4  +  vTTvT+l  =  Vl4T"vf+l 

=  V14  +  2  =  4. 


RADICALS  271 


4.   Given  ^V^Zl^  =  SV^-2b^  ^^  ^^^  ^^^  ^^^^  ^^  ^^ 


2  Vaa;  +  b     SVax  +  36 
Solution 

2Vax-b  _SVax-2b 
2Vax+h     3Vax  +  36 

Reducing  to  mixed  numbers,  1 ^^ =  1 


Canceling, 
Dividing  by  —  6, 


2Vax-^b  SVax  +  Sb 

2  b ^  _         6b 

2y/ax  +  b         SVax-\-Sb 
2  5 


2Vax-\-b     SVax  +  Sb 
Clearing  of  fractions,  etc.,  6  Vox  —  10  Vox  =  56-66. 

4 

Squaring,  etc.,  x  =  — • 

Suggestions.  —  1.  When  the  equation  is  free  from  fractions, 
transpose  so  that  the  radical  term,  if  there  is  but  one,  or  the  more 
complex  radical  term,  if  there  is  more  than  one,  may  constitute  one 
member  of  the  equation ;  then  raise  each  member  to  a  power  of  the 
same  degree  as  that  radical.  Simplify  the  result.  If  the  equation 
is  not  freed  from  radicals  by  the  first  involution,  proceed  again  as 
at  first. 

2.  It  is  sometimes  convenient  to  rationalize  denominators  before 
clearing  of  fractions  or  involving. 

Solve  the  following  equations : 


5.  Va;  +  ll=4^  12.  l-{-2Vx  =  7  —Vx. 

6.  Va;  +  5  =  3.  13.  ■Vx-\-16-Vx  =  2. 

7.  \^x-a^=  b.  14.  V2^-V2a;-15  =  l. 

8.  -J/^^^  =  2.  15.  V^T^Tl  =  2-a;. 

9.  -y/x^^^^a.  16.  3 V^^^  =  3 .t - 3. 

10.  ^x-{-b  =  a.  17.  V^  +  2=Va;+32. 

11.  l-hVa  =  6.  18.  5-V^T^5=V«. 


272 

19.    2Vx—x. 


ACADEMIC  ALGEBRA 

x—S^/x.  25.    V3 x  —  5  +  V3 05  +  7  =  6. 


20.    V4»2-|-6aj-10  =  2a;+4.         26.    Vl6  x-\-S-\-  V16  x  +  8  =  5. 

27.  V9x  +  8+V9^-4  =  0. 

28.  VlT^A^Tl^  =  1  +  aj. 
4  =  0. 

24.    V2x-l+V2a;  +  4=:5.      30.    2a;+V4^^w!^^^  =  l. 

31.   V7  +  Vl  +^4  +  \'H-  2  V^  =  3. 


21.  Var^  —  5a;  +  7  +  2  =  a;. 

22.  4-V4-8x4-9a;2  =  3a;. 

23.  V2(l-ic)(3-2a;)-l=2aj.    29.    A/7+3\/5^^f^ 


33. 


34. 


35. 


^36. 


37. 


32.    ^ 

V3a; 

+  2 

V2^  +  9 

_V2a;-h20 

V2^-7 

V2^- 

12 

V^  +  18 

V^  +  2 

32 

V^  +  6 

+  1 

Va;-1 

Va;-3 

Vaj4-5 

Va;-1 

V5-6 

V^-8 

V^-1 

V^-5 

Va;-3 

Vir-4 

=  V3a;  +  2+V3a;-l. 

2^4-6      V2ic  +  2 


38. 


39. 


40. 


41. 


^  +  4      V2ic  +  1 


llx  +  V2 a;  +  3  ^ 8 
U^-V2a;  +  3     3 


2V2a;4-4     3Va;4-l4-9 


2V2a;-4      3Va;  +  l-9 

^iV5x-9  ^'VV5^-21_ 


VaJ  +  1      Va;  —  2 
43 


42     V4a;  +  34-2Va;-l^g 
V4a;  +  3-2Vx^^ 


Va;4-1— Va;— 1^1^ 
Va;  +  1  +  Va;  —  1     2 


44. 


45. 


^-^_  =  -^+^^  +  2V3. 
V^-V3  2 


19a;4-V2a;  +  ll 


2i. 


Vl9a;-V2a;  +  ll 
46.   2V^-V4a;-22-V2  =  0. 


RADICALS  273 


47.    H-V(3-5a;)2  +  16  =  2(3-a;). 


48.  Vx  +  Vaj  — Va-  — x  =  Va. 

49.  V^  +  Vaj-(a-6)2  =  aH-6. 


50.  Vmn  —  x  —  Va?  Vmri  —  1  =  Vm?i  VT 

51.  a-\/x  —  hVx  =  a- -{- Ir  —  2  ah. 


.   ^52.    V5  aa;  —  9  a^  -f-  a  =  V5  oa;. 
6  a 


63.    V«  +  3a  = ^^^^^^ Va;. 

Va;4-3a 

54.    V2x-V2a;-7  = i- 


V2a;-7 
55.    V2a;+Vl0ifH-l=V2^  +  l. 


Va;  +  a  —  Va;  —  a  ^ 

67.    V»  +  V2^  +  V3^  =  Va. 

Solution 

\^  +  yftx  4- \/8x  =  Va. 

Factoring,  Vx(l  +  V2  +  VS)  =  Va. 

Multiplying  by  1  +  \/2  —  Vii, 

>/i(l+2V2  +  2-3)  =  v^(l  +V2-V3). 

Vx  .  2  >/2  =  Va(l  +  \/2  -  V3). 

Squaring,  '  8a;  =  a(l+\/2-  V3)2. 

a;  =  ^(l +V2-v^3)2. 
8 

58.  V2«  +  V3«  +  V5^  =  Vm. 

59.  v'2^4-V3lc-V5^  =  Vc. 


^ 


V 60.    ViC— «  +  V2 (a;  —  a)  =  A/3 a;  +  a V2. 
A  61.    Va;-l-|-V2a;-2  =  V3a;-3H-V2. 

62.    V2a;-3  +  V4a;-6  =  V2^  +  V^. 


ACAD.    ALG. — 18 


274  ACADEMIC  ALGEBRA 

REVIEW 

Reduce  the  following  to  their  simplest  forms : 

-     6  a^  —  7  0/*^  —  5  a?^  x  —  y     y  -\- x      ^x^y^ 

9a^  —  25a;  '    x-\-y     y  —  x     x^  —  y^ 

2     8a;^  +  18a;-5  ^     V2-V3      V2+V3 

•    l2a;2  +  5a;-2  '    V2+V3      V2-V3' 

^     a^:t?  —  aVa; -}-  a?^  x-VyJxy^y  ,  xVx-[-yVy 


4. 


■^^  "*     Vaj  +  V?/  ^  +  2/ 

a2-2aV6  +  6        •  1  1 
10.                  _-l  + 


«-V6  *    Va+V6  Va-V6 

V2+V3  4-V5  '  vrT^+vr=^ 

2+V5      V243  1-V2a;    1  +  V2a;    l-2a; 

13     a^  +  Var^  —  g^  a;  —  Vx^  —  a^  ^ 

X  —  Va;^  —  a^  a;  4-  Va;^  —  (^ 


14. 


V a  +  1  4-  Va  —  1      A/a  4-  1  —  V a  —  1 


Va  +  1  —  Va  —  1      Va  4- 1  +  Vtt  —  1 


15      a'-2aa;-3a;^       6  a^  +  7  aa;4- 2  a;^ 
3a2  4_5aa;4-2a;2       a2-4aa;4-3a:2' 

1.  a2-6  a2-4aV5  4-4  6 

lb. X 


a^-  2  aV6  4-  &       a'  +  2  aV6  +  6 


/6\       Va& 
a  +  6' 


1  +  g  4-  g'^  /  g        Va;\/  a V^\ 

14-Va  +  a  „^     W^       a  Ay^_«J 

~ 7"= *  20.     -— r= — • 

1  —  Va  4-g  /  g   _  -yx\  f  a   _  Va;\ 


REVIEW  275 

Expand : 

21.  (a-'-b'f.  25.    (a-2  +  a-i)2.  31.    (a-Vly. 

22.  C2a-3by.  ''•    (^"  +  ^)*-  32.    (Vx+V^)«. 

27.    (a^-6^)«.  33.    (V2-V3/. 

^^'    (I  "2)*  28.    (a^-6-i)^  34.    (V5-2)«. 

29.    (a-^-6-^)«.  35.    (^/4-^/. 

24.    [ax  +  -y  ^^     (a^+6^)«.  36.    (V2-^/2)«. 

Extract  the  square  root  of 

37.  ?|!  +  3ar^-a5^-^  +  l 

38.  ^'  +  42/^+:^-2ar2/4--^-?/2^. 
4  16  4 

39.  a'  +  12ah^  +  54.ab  +  10Sah^  -\-Slb\ 

40.  1 +2V^  — «  — 2a;V«  +  ar^. 

41.  a-\-4.b  +  9c  — Wab-\- 6 Vac  — 12 Vbc. 

42.  x"  —  4:xVxy  -\-6ccy  —  4:  yVxy  +  ^^ 

43.  a^-12  icV  +  60  x^y  -  160  ic?/^  +  240  rcV 

-192a;V  +  64  2/3^ 
Find  the  square  root  of 

44.  81234169.  48.   56  +  14 Vl5. 

45.  64064016.         '  49.   47  - 12  Vl5. 

46.  .00022801.  60.    62  +  20v'6. 

47.  .1  to  four  places.  51.   51  —  36V2. 

Extract  the  cube  root  of 

52.  a^-9a;+27a;-^-27a;-3. 

53.  27^  +  27^-5+^-.^. 

54.  a:^-^3x^Vx  —  5xVx-}-3Vx  —  l, 


276  ACADEMIC  ALGEBRA 

65.  Find  the  cube  root  of  2  V2  -  6^2  +  3  V2  -y/l  -  2. 

56.  Find  the  cube  root  of  (a  +  12  6  -f  3  c)Va 

-(6a-\-Sb-\-(y  c)-\/h-  (3  a  +  12  6  +  c)^c  +  12  Va6c. 

67.  Extract  the  cube  root  of  510,082,399. 

58.  Extract  the  cube  root  of  1,042,590,744. 

59.  Extract  the  cube  root  of  2  to  three  decimal  places. 


60.  Find  the  first  four  terms  of  Vl  -^x  —  xK 

61.  Find  the  first  three  terms  of  Vl  -f-  a^. 

62.  Find  the  fourth  root  of 

a«  -  4  a'A/W^  +  6  a^h-^  -  4  a^-^ Va6=i  +  h-\ 

63.  Find  the  sixth  root  of 

8  -  48  Va  +  120  a  -  160  a■^/a  +  120  a^  -  48  aVa  +  8  aK 

If  a"*  X  a"  —  a"*+"  for  all  values  of  m  and  n,  show  that 

64.  a^  =  — •  ^     ^ 

a- 

65.  a^=V^3  =  (V5)».  ««•    W  =  «W- 

66.  2.-i  =  2j^.  ■  69.    f|Y=^. 

Find  the  values  of  the  following : 
^0.    16l  73.    (aV)i  76.    (^)"^. 

71.  27i  74.    (6y)"l       •  77.    (36)-| 

72.  8-*  76.    (a''6")~".  78.    (-^V)"*- 

79.  For  what  values  of  n  is  (a  —  6)**  =(h  —  aY? 

Simplify  and  express  with  positive  exponents : 

80.  (36  a-»  ^  25  a-2)-l  83.    ( VaV^^ -- VaV^^)^ 

81.  (8  a»a;^  X  64  a-^aj-^)"^.  84.    (Vcr^^-h  Vo^ft)"^. 

82.  iah^)^  ^  {ah^y.  85.    ( Va  ^  Va) -r- Va. 


REVIEW 


277 


86. 


a3  —  &3      a^-^  63 


a-{-b         2  ah 


a3 


bi 


1  +  a-^6  .  n  +  a6-^  +  a-^5-^     1  +  a  'b^ 
1  -  a-ift  '  \1  -  a6-i  +  d'b-'-     1  -  a-36V' 


—  a6-^  +  d^b- 

Solve  the  following  equations : 

X  -^1      x  —  1      S  —  5x 


88. 


89. 


90. 


91. 


92. 


93. 


x-1      x-\-l       1-a? 

l-2x     2a;-l      ,     ^5a;-6JL     17 +3  a; 
10  5  ^'^  2  a;  30      ' 

4a;-17     3|-22a;^^  _  ?  A  _  ^A 
9  33  ^     a;V       W* 

'Sx-5y     2x-Sy-9^y      7  ^ 
3  12  2     12* 

J(M-H)-f(--|-24)  =  0. 

r3a;-M=2y, 

l(a:  +  5)(2/  +  7)  =  (a;+l)(2/-9)  +  112. 

x-y  =  Sy 

(x-\-l)(x  +  2)-(x-2)(x-\-l)  =  lly-\-2. 


Simplify  and  express  with  positive  exponents : 


94. 


V27  •  tJ  ''•  L^-^(^J  • 


3  a-1  +  2  a; 


95.    fa-2[a^(a^)*]»sl 

(: 


98.    \(ah^)^^(ar^b)-^K 
a-\-b         a—b 


96. 


6 


99 


100. 


V-  — '   . 


b  +  a 
ab 


QUADRATIC   EQUATIONS 


285.  1.  What  is  the  value  of  x  in  the  equation  3x  =  24? 
What  kind  of  an  equation  is  it  ? 

2.  What  powers  of  x  are  found  in  the  equation  x^  -\-2x  =  3? 
Which  is  the  higher  power  ? 

3.  What  is  the  value  of  x  in  the  equation  x"'  =  9?  How  many 
values  has  x?     How  do  they  compare  numerically  ? 

4.  Factor  x^  —  5x-{-6  =  0,  and  so  find  the  values  of  x.  How 
many  values  has  x  ? 

''^  286.  An  integral  equation  that  contains  the  square  of  the 
unknown  number,  but  no  higher  power,  is  called  a  Quadratic 
Equation,  or  an  equation  of  the  second  degree. 

It  is  evident,  therefore,  that  quadratic  equations  may  be  of 
two  kinds  —  those  which  contain  only  the  second  power  of  the 
unknown  number,  and  those  which  contain  both  the  second  and 
first  powers. 

a;2  =  15  and  aa;'-^  -^hx  =  c  are^ quadratic  equations. 

PURE  QUADRATICS 

287.  An  equation  that  contains  only  the  second  power  of  the 
unknown  number  is  called  a  Pure  Quadratic. 

ax2  =  h  and  ax"^  —  cx'^  =  he  are  pure  quadratics. 

Pure  Quadratics  are  also  called  Incomplete  Quadratics,  because 
they  lack  the  first  power  of  the  unknown  number. 

288.  Since  pure  quadratics  contain  only  the  second  power  of 
the  unknown  number,  they  may  be  reduced  to  the  general  form 
ax^  =  h,  in  which  a  represents  the  coefficient  of  a^,  and  h  the  sum 
of  the  terms  that  do  not  involve  x^. 

278 


QUADRATIC   EQUATIONS  279 

289.  Principles.  —  1.  If  the  square  root  of  each  member  of  a 
quadratic  equation  is  extracted  and  the  second  member  of  the  result- 
ing equation  is  given  the  sign  ± ,  the  resulting  equation  is  equivalent 
to  the  given  equation. 

2.  Every  pure  quadratic  equation  has  two  roots  numerically  equal, 
but  having  opposite  signs. 

For  the  equation  A^  =  :^,  or  A^  -  B^  =  0,  or  {A  -  B)(A+  B)=  0,  is 
equivalent  to  the  two  equations 

A-  B  =  0  a,nd  A-\-  B  =  0, 

or  A=+  B  and  A=-  B; 

that  is,  to  the  two  equations  A  =  ±  B. 

Examples 

1.  Given  10  a:^  =  99  —  a:^,  to  find  the  value  of  x. 

Solution 
10x2  =  99 -x2. 
Transposing,  etc.,  11  a;^  =  99. 

Dividing  by  11,  x^  =  9. 

Extracting  the  square  root  of  each  member,  Ax.  7, 

a;=±3. 

Strictly  speaking,  the  last  equation  should  be  ±x  =±S,  which  stands  for 
the  equations  -fx=+3,  4-x  =  —  3,  — x  =  —  3,  and  —x=+S.  But  since 
the  last  two  equations  may  be  derived  from  the  first  two,  the  first  two  express 
all  the  values  of  x.  For  convenience,  the  two  expressions,  x  =  +  3  and 
X  =  —  3,  are  written  x  =  ±  3. 

Consequently,  in  extracting  the  square  roots  of  the  members  of  an  equa- 
tion, it  will  be  suflBcient  to  write  the  ambiguous  sign  before  the  root  of  one 
member. 

2.  Find  the  roots  of  the  equation  3  ic^  =  — 15. 

Solution 
3x2  =-15. 
Dividing  by  3,  x2  =  -  5. 

Extracting  the  square  root,        x  =  ±  V  —  5. 


280  QUADRATIC  EQUATIONS 

Solve  the  following  equations : 

3.  7a^- 25  =  50^  +  73. 

4.  (aj  +  4)2  =  8«  +  25. 
6.    {a-xf=(3x-\-a){x-a). 

6.  ax'={a-h){a?-h^)-hx'. 

7.  0^x^+2 ax'={(i'-lY-a?. 

8.  (a^  +  2)2-4(a;  +  2)  =  4.        17.    V^?+8 ^       =a?. 

Vaj2  +  8 

9.  ^^  =  _^.  ,«     ^  ,  ^/^  ,  ^2_       2m^ 


14. 

x 

a-h 

X 

=  0. 

a  +  h 

1  *! 

x  —  3     ic-f-3 

=  H. 

x-2  ' 

'  aj  +  2 

16. 

X 

"6~~^T8*  1^-    a;+Va:^  +  m2  = 


10.    -^^ 1 ?— =  2|.  19     a;  +  g      x—a^a^^})^ 

1-x      l  +  a;  •    a;+6'^a;-6      a^-d^* 

-  -      a;,  a^  — 15     a;  ^^ — 7^  6 


7)1^ 


12         5a;         5 


Va^'  +  S 


12.  ^±^  +  ^z^  =  4.  21.    _J4 v^:n2  =  a:. 

x-3     x  +  S  Var»  + 12 

13.  ^^L?  _L.  ^±_5  =  _  1  22     ^_±_^_|_^  — «        2<* 


aj  +  1     a;  —  1  x  —  a     x  +  a     1  —a 

a;  +  7  aj  —  7  7 


23. 


a;2_7a;     a^  +  7a;      a;2-73 


24.    x-\-V^ 


Va^  —  a^ 


25.    V25-6a;  +  V25H-6a;  =  8. 


26^    2x±V±^-l^^ 

2a;-V4a^-l 

27.  _vr+^    _VlH^=0. 


1+Vl  +  a;     1+Vl  — a; 

Va^-f  l-Va;2- 

-1      1 

Var^-hl+Va:2- 
l 

ri     2 

1 

28.        

Var^-hl+Va:'-!      ^ 

29.  ^  ^  1 


<-> 


1-l-Vl-a;     vT-hx-hl     ^ 


QUADRATIC   EQUATIONS  281 


2  2 

30.    =^:^^rH =a?. 

a;  +  V2  —  ic^     a:  —  V2  —  ar^ 


31. 


Va;  -t-  2  g  —  V  a;  —  2  a  __  a; 
Va;-2a  +  Va;4-2a     2a 


32.    J^^+\/^^  =  «^- 
^a;  +  a      ^/a;  —  a 


Problems 

1.  If  25  is  added  to  the  square  of  a  certain  number,  the  sum 
is  equal  to  the  square  of  13.     What  is  the  number  ? 

2.  What  number  is  that  whose  square  is  equal  to  the  differ- 
ence of  the  squares  of  25  and  20  ? 

3.  If  a  certain  number  is  increased  by  5  and  also  decreased 
by  5,  the  product  of  these  results  will  be  75.  What  is  the 
number  ? 

4.  Two  numbers  are  as  3  to  4,  and  the  sum  of  their  squares  is 
equal  to  the  square  of  15.     What  are  the  numbers  ? 

5.  Two  numbers  are  as  4  to  3,  and  the  sum  of  their  squares  is 
400.    What  are  the  numbers  ? 

6.  A  gentleman  has  two  square  rooms  whose  sides  are  as 
2  to  3.  He  finds  that  it  takes  9  square  yards  more  than  twice  as 
much  carpeting  for  the  larger  room  as  for  the  smaller.  What  is 
the  length  of  a  side  of  each  room  ? 

7.  A  man  who  owns  a  field  80  rods  square  sells  one  fourth  of  it. 
If  the  part  he  sells  is  also  a  square,  how  long  is  each  of  its  sides  ? 

8.  A  man  had  a  rectangular  field  whose  width  was  f  of  its 
length.  He  built  a  fence  across  it  so  that  one  of  the  two  parts 
thus  formed  was  a  square.  .  If  the  square  field  contained  10  acres, 
what  were  the  dimensions  of  the  original  field  ? 

9.  How  many  rods  of  fence  will  inclose  a  square  garden 
whose  area  is  2\  acres  ? 

10.  The  sum  of  two  numbers  is  10,  and  their  product  is  21. 
What  are  the  numbers  ? 

Suggestion.  —  Represent  the  numbers  by  5  -f  x  and  5  —  OJ. 


282  ACADEMIC  ALGEBRA 

11.  The  sum  of  two  numbers  is  16,  and  their  product  is  55 
What  are  the  numbers  ? 

12.  The  sum  of  two  numbers  is  26,  and  their  product  is  69. 
What  are  the  numbers  ? 

13.  The  sum  of  two  numbers  is  5,  and  their  product  is  — 14. 
What  are  the  numbers  ? 

''^14.   Factor   a^  +  17a  +  60   by   the   method   suggested   in   the 
preceding  problems. 

Suggestion,  -f  60  is  the  product  of  the  arithmetical  terms,  and  +  17  is 
their  algebraic  sum. 

15.  Separate  a^  +  2a  —  2  into  two  factors. 

16.  Separate  oc^  —  2x—l  into  two  factors. 

17.  Divide  24  into  two  parts  whose  product  is  143. 

18.  The  length  of  a  ten-acre  field  was  4  times  its  width.  What 
were  its  dimensions  ? 

19.  The  sum  of  the  squares  of  two  numbers  is  394,  and  the 
difference  of  their  squares  is  56,     What  are  the  numbers  ? 

2(k  A  man  has  two  square  fields  that  together  contain  51^ 
acres.  If  the  side  of  one  is  as  much  longer  than  50  rods  as  that 
of  the  other  is  shorter  than  50  rods,  what  are  the  dimensions  of 
each  field  ? 

AFFECTED   QUADRATICS 

N  290.  A  quadratic  equation  that  contains  both  the  second  and 
the  first  powers  of  one  unknown  number  is  called  an  Affected 
Quadratic. 

jr2  +  3  aj  =  10,  i  x^  —  X  +  1  =  0,  and  ax^  +  6ic  -f  c  =  0  are  affected  quadratics. 
Affected  Quadratics  are  also  called  Complete  Quadratics. 

291.  Since  affected  quadratic  equations  contain  both  the  second 
and  first  powers  of  the  unknown  number,  they  may  always  be 
reduced  to  the  general  form  of  aoc^  -f  6a;  +  c  =  0,  in  which  a,  b, 
and  c  may  represent  any  numbers  whatever. 

The  term  c  is  sometimes  called  the  absolute  term. 


QUADRATIC  EQUATIONS  283 

292.    To  solve  affected  quadratics  by  factoring. 

Reduce  the  equations  to  the  form  ax^  +  do?  +  c  =  0,  and  solve  by 
the  methods  of  §  141. 

Solve  the  following  equations  by  factoring : 

1.  0.-2-50^  +  6  =  0.  ^7.   10a^-27a;  +  5  =  0. 

2.  a52-5x  =  24.  8.   Q{x'^l)=13x. 
^3.   a^-l  =  3(a;  +  l).                ''S.   a^  -  {a  -  h)x  =  ah. 

4.  2a^-7a:  +  3  =  0.  •*'lO.    2^ -^ax-2  a'  =  (). 

^5.  2  a^ -a; -3  =  0.  -^11.    3(b^ +  x')=10hx. 

6.  3aj2-2iB-8  =  0.  12.    a^-2aa;  +  (a  +  l)(a-l)  =0. 

13.  Solve  the  equation  a:^  _|_  loo  x  +  2491  =  0. 

Solution 

Since,  §  99,  100  Is  the  sum  of  the  arithmetical  terms  of  the  factors  ot 
a;2  +  100  X  +  2491,  and  their  product  is  2491,  60  +p  and  50  —p  may  repre- 
sent the  two  factors  of  2491  whose  sum  is  100. 

Then,  (50  +  p) (50  -p)=  2491. 

Expanding,  2500  -p'^  =  2491. 

Transposing,  etc.,  p2  _  9, 

p  =  ±  3. 
50  +  p  =  53,   50  -  p  =  47. 

Therefore,  §  130,  a;2  +  100  x  +  2491  =  (x  +  53)  (x  +  47)  =  0, 

and  X  =  —  53  or  —  47. 

Since  |)  =  —  3  gives  no  new  values  of  50  +  p  and  50  —  p,  the  negative  value 
of  p  may  be  disregarded. 

14.  Solve  the  equation  a.-^  -h  3  a;  -  208  =  0. 


Solution 

x2  +  3x-208  = 

0. 

Let 

(f+p)(!-JP)  = 

:  -  208. 

Expandin 

g.                                        f-p^  = 

:  -  208. 

Solving, 

P  = 

=  ±¥. 

Factoring 

the  given  equation, 

(x  +  16)(x-13)  = 

:16,     f- 
:0. 

p  =  - 

13. 

•*• 

i                               X  = 

:  -  16  or 

+  13. 

284  ACADEMIC  ALGEBRA 

Solve  the  following  equations : 

15.  a;2  _|_  10  ^^.  _|_  21  =  0.  28.  x^ -\-x-756  =  0. 

16.  a^  +  12a;-28  =  0.  29.  a;^  _  ^^  _  506  =  0. 

17.  aj2- 20  a; +  51  =  0.  30.  a;^  ^2  a?  -  168  =  0. 

18.  a?2  +  60a;  +  891  =  0.  31.  a;^  +  6  a;  -  135  =  0. 
•^19.  a;2-44a;  +  403  =  0.  32.  a;^  +  3  a;- 154  =  0. 
^20.  a^  +  20a;-629  =  0.  33.  a^  +  5  a;  +  2  =  0. 
>i21.  a;2- 30  a; -2275  =  0.  34.  0^2  _^  a;  -  10  =  0. 

22.  a;2  +  24a;  +  119  =  0.  35.  a;^  -  a;  -  1  =  0. 

23.  a^  + 2  a; -323  =  0.  36.  aj2-2a;-4  =  0. 

24.  a^-6a;-475  =  0.  37.  a;2-3a;-9  =  0. 

25.  a^  + 8  a; -768  =  0.  38.  a^  +  4a;  +  8  =  0. 

26.  a^  + 3  a; -418  =  0.  39.  a;^ -f  6  a;  +  14  =  0. 

27.  x'-^-Bx- 5^6  =  0.  40.  a;^ -f- 8  a;  =  - 25. 

293.   First  method  of  completing  the  square. 

1.  What  is  the  square  of  a;  +  3  ?   of  a;  +  5?   ofa;  +  10? 

2.  If  a^  +  20  a;  are  the  first  two  terms  of  the  square  of  a  bino- 
mial, what  is  the  first  term  of  the  binomial-? 

3.  Since  20  a;  is  twice  the  product  of  the  two  terms  of  the 
binomial,  and  the  first  term  of  the  binomial  is  a?,  how  may  the 
second  term  of  the  binomial  be  found  ? 

4.  Since  the  second  term  of  the  binomial  is  10,  what  must  be 
added  to  a.-^  +  20  a;  to  complete  the  square  of  the  binomial  ? 

5.  What  term  must  be  added  to  x^-{-6x  to  complete  the 
square  of  some  binomial?     How  is  the  term  found? 

6.  What  term  must  be  added  to  a^  +  Sx  to  complete  the 
square  ?    to  a;^  — 10  a;  ?    to  a;^  —  14  a;  ? 

7.  What  must  be  added  to  both  members  of  the  equation 
ar*  —  12  a?  =  13  to  make  the  first  member  a  perfect  square  ? 


QUADRATIC  EQUATIONS  285 


Examples 

1.  Solve  the  equation  ic^  —  6  a;  =  40. 

PROCESS  Explanation.  —  Completing   the    square   in   the 

first  member  by  adding  lo  each  member  the  square 
ar  —  6 a;  =  40        ^f  j^^jf  ^^^  coefficient  oi  x,  x^  -Qx  +'9  =  49. 
a:^-- 6  a; +  9  =  49  Extracting    the    square    root   of    each    member. 

'a;-3  =  ±7     ^-3=±7. 

Using  first  the  upper  sign  of  ±  7,  the  simple  equa- 
x  =  -\-l-\-o  =  i.y)        tion  X  —  3  =  +  7  gives  x  =  10.     Next  using  the  lower 
iC  =  _7-[-3  =  _4      sign  of  db  7,  the  simple  equation  x  —  3  =  —  7  gives 
x  =  -4. 

Since  each  of  the  values  10  and  —  4  satisfies  the  given  equation  when 
substituted  for  x,  x  =  10  or  —  4. 

2.  Solve  the  equation  a;^  —  5  a;  =  14. 

Solution 

x2-5x  =  14. 
Completing  the  square,  x^  —  5  x  +  (f  )2  =  14  +  ^  =  -S^. 

Extracting  the  square  root,  x  —  ^  =  ±^. 

Taking  the  upper  sign,  x  =  ^  +  f  =  7. 

Taking  the  lower  sign,  x  =  f  —  f  =  —  2. 

3.  Solve  the  equation   10  a:^  +  19  a;  =  15. 

Solution 

10x2  +  19x  =  15. 
Dividing  by  coefficient  of  x^,  x^  +  |^  x  =  |^. 

Completing  the  square,  x^  +  j|  x  +  (M)'  =  i^  +  IM  =  IM- 

Extracting  the  square  root,  x+  h^=±ih- 

Taking  the  upper  sign,  a:  =  -  ^  +  f  ^  =  f . 

Taking  the  lower  sign,  x  =  -  ^^  -  f ^  =  -  f . 

Rule.  —  Transpose  so  that  the  terms  containing  ^  and  x  are  in 
one  member  of  the  equation  arid  the  known  terms  in  the  other,  and 
make  the  coefficient  of  x^  unity  by  dividing  both  members  of  the 
equation  by  the  coefficient  of  x^. 

Add  to  each  member  of  the  equation  the  square  of  half  the  coeffi- 
cient of  X,  and  extract  the  square  root  of  each  member. 

Solve  the  two  simple  equations  thus  obtained. 


286  ACADEMIC  ALGEBRA 

Find  the  values  of  x  in  the  following  equations : 

4.  x^-2x  =  US. 

5.  x^-\-2x=ieS. 

6.  ar'-i^-llT. 
.  7.    a^-6a;=160. 

8.  ic2_8a;=180. 

^  9.  a;2  +  2a;  =  120. 

10.  aj2  +  6a;  =  187. 

^11.  x2- 12  a;  =  189 

12.  i«2  +  10a;  =  171. 

13.  a;2-22a;  =  48. 
^14.  a;2  +  30a;  =  31. 

26. 
27. 

28.  ^  . 

x-\-2         X  2x 

29.  a;^  +  (m  +  n)  (m  —  n)  =  2  7^a;. 

294.   Other  methods  of  completing  the  square. 

By  the  previous  method,  when  x^  had  a  coefficient,  the  equation 
was  divided  by  that  coefficient  so  that  the  ?erm  containing  xr  might 
always  he  a  perfect  square.  The  same  result  may  be  secured  in 
other  ways. 

Thus,  if  the  term  containing  x^  is  3  oi?,  it  may  be  made  a  perfect 
square  by  multiplying  by  3 ;  if  8  x^,  by  multiplying  by  2 ;  if  ax^^ 
by  multiplying  by  a. 

In  the  completed  square  aV  +  2  dbx  4-  h^,  it  is  evident  that  the 
third  term,  6^,  is  the  square  of  the  quotient  obtained  by  dividing 
the  second  term  by  twice  the  square  root  of  the  first 


15. 

a^  +  3aj  =  10. 

16. 

a^- 3a;  =  180. 

17. 

a;2-}-15a;  =  54. 

18. 

x'-x=  930. 

a9. 

x^  +  13  a;  =  140. 

^J20. 

a^_llaj4.28  =  0. 

^21. 

5a;2_3^_2  =  o. 

22. 

6aj2-5a;-6  =  0. 

23. 

2x2  + 9a;  =  35. 

24. 

3ar^-7a;  =  10. 

25. 

4a;2-19a;  =  5. 

13          10 
a,  +  l      x-1       3 

a?        3a;- 5 

x  +  2 

a;-2           2 

5 

1         a;-2_a;-7. 

QUADRATIC  EQUATIONS  287 

Examples 

1.  Solve  the  equation  5  a^  +  12  a;  =  9. 

Solution 
5x2 +  12  a;  =  9. 
Multiplying  by  6,  25  x^  +  60  x  =  45. 

Completing  the  square,  25  x^  +  60  x  +  36  =  81. 
Extracting  the  square  root,  6  x  -f  6  =  i  9. 

5  X  =  -  6  ±  9. 
.-.  X  =  f  or  -  3. 

Explanation.  — Since  the  coefficient  of  x^  is  not  a  perfect  square,  it  may 
be  made  a  perfect  square  by  multiplying  the  members  of  the  equation  by  5. 

Since,  if  to  25  x^  +  60  x  there  were  added  such  a  term  as  would  make  the 
trinomial  a  perfect  square,  60  x  would  be  equal  to  twice  the  product  of  the 
square  root  of  25  x^  and  the  square  root  of  this  third  term,  the  square  root 
of  the  third  term  is  obtained  by  dividing  60 x  by  2>/25x^  ;  that  is,  by  10 x. 
60  X  -7-  10  X  =  6,  and  6^,  or  36,  added  to  both  members  completes  the  square 
of(5x  +  6). 

2.  Solve  the  equation  8  a^  —  10  a;  =  3. 

Solution 
8x2-10x  =  3. 
Multiplying  by  2,  16  x2  -  20  x  =  6. 

Completing  the  square,  16  x2  -  20  x  +  (f  )2  =  6  +  ^  =  ^. 
Extracting  the  square  root,  4  x  —  f  =  ±  |. 

'*a;  =  f±^  =  6  or  -1. 
.-.  X  =  f  or  —  J. 

General  Rule.  —  Transpose  so  that  the  terms  containing  o^  and  x 
are  in  one  member  of  the  equation  and  the  known  terms  in  the  other. 

If  the  term  containing  the  second  power  of  the  unknown  number  is 
not  a  perfect  square,  make  it  such  by  multiplying  or  dividing  the 
members  of  the  equation  by  some  number. 

Add  to  each  member  of  the  equation  the  square  of  the  qxLotient 
obtained  by  dividing  the  term  containing  the  first  power  of  the 
unknown  number  by  twice  the  square  root  of  the  term  containing  the 
second  power. 

Extract  the  square  root  of  each  member,  and  solve  the  two  simple 
equations. 


288  ACADEMIC  ALGEBRA 


Solve  tne  following  equations : 

3.   2a^-5aj  =  42. 

8.   3ar^-|-4aj  =  95. 

4.    6x'-5x-\-l  =  0, 

9.    7x'  +  2x  =  S2. 

5.   4ar^-12a;  =  27. 

10.   8-ar'-18a;  =  5. 

6.   8ar^  +  20a;  =  48. 

11.   6«2_|_5^^4 

7.    lSx'-^6x  =  4:. 

12.    5ar^4-6a;  =  8. 

13.    Solve  the  equation 

aar*  +  6a;  4-  c  =  0. 

' 

Solution 

ax^-\-bx-h  c  =  0. 

(1) 

Transposing  c, 

aa;^  +  ^)x  =  -  c. 

(2) 

Multiplying  by  a, 

a'^x^  +  a6x  =  —  ac. 

(3) 

Completing  the  square, 

a^'i  +  abx  +  ^  =  ^-ac. 

(4) 

Multiplying  by  4, 

4  a2a;2  +  4  a6a;  +  62  =  62  _  4  ^c. 

(5) 

Extracting  the  square  root. 

2ax  +  b  =  ±  Vb'^  -  4  ac. 

(6) 

.  „      -b±y/b'^-4aG 

(7) 

2a 

It  is  evident  that  (5)  can  be  obtained  by  multiplying  (2)  by  4  «  and  add- 
ing 62  to  both  members.  Hence,  when  a  quadratic  has  the  general  form  of 
(1),  if  the  absolute  term  is  transposed  to  the  second  member,  as  in  (2),  the 
square  may  be  completed  and  fractions  avoided  by  multiplying  by  4  times  the 
coefficient  ofx^  and  adding  to  each  member  the  square  of  the  coefficient  ofx  in 
the  given  equation. 

This  is  called  the  Hindoo  method  of  completing  the  square. 

Solve  the  following  equations  by  the  Hindoo  method : 

14.  2«24-3a;  =  27.  21.  4.x^-x-S  =  0. 

15.  2a;2  +  5a;  =  7.  22.  5a^-2x-16  =  0. 

16.  2iB2  +  7a;  =  -6.  23.  3a^  + 7a;- 110  =  0. 

17.  Sx'-5x  =  2.  24.  2ar'-5.T-150  =  0. 

18.  4ar^-15a;  =  4.  25.  3a;2  +  a;- 200  =  0. 

19.  5ar'-7a;  =  -2.  26.  ISar^- 7a;- 2  =  0. 

20.  6ar'  +  5a;  =  -l.  27.  7a;2_  20a;_  32  =  0. 


QUADRATIC  EQUATIONS  289 

295.   To  solve  quadratics  by  a  formula. 

Since  every  quadratic  can  be  reduced  to  the  general  form 
ax^  -{-hx-{-  c  =  0,  in  which  a,  h,  and  c  represent  any  numbers 
whatever,  and  since  the  roots  of  this  equation  are 

Ex.  13,  §  294,  X  =  -6±V6^-4ac^ 

the  values  of  the  unknown  number  in  any  affected  quadratic 
equation  may  be  found  by  substituting  the  coefficient  of  ^  for  a 
the  coefficient  of  x  for  h,  and  the  absolute  term  for  c. 

Examples 
1.    Solve  the  equation  6  a^  —  a;  — 15  =  0. 
Solution.  —  Since  a  =  Q,  6  =—  1,  and  c  =  —  15,  by  the  above  formula, 


l±V(^iy-^-4  x6(-15) 
2x6 


12         3  2 

Solve  by  the  above  formula : 

2.  2a^-f5a;  +  2  =  0.  10.  2a^  +  3aT-l  =  0. 

3.  3ar^  +  lla;  +  6  =  0.  11.  3ic*  +  2a;- 4  =  0. 

4.  6ar^-7x  +  2  =  0.  12.  r^-5a;  =  -3. 

5.  4ar'  +  4a;-15  =  0.  13.  ^x^-Qx  =  -2. 
Oj6.   2ar^  +  3a;-9  =  0.     "  14.  4ar'- 3a;- 2  =  0. 

7.    2ar'H-3a;H-l=0.  15.  a?  -  Qx-\-10  =  0.^' 

^^  8.    3a;2-13a;  =  10.     -^  16.  a;^  +  4  a- _^  12  =  0. ' 

9.    7ar'  +  9x  =  10.  17.  x*-8a;  =  -20. 

Solve  by  any  method : 

"^8.    a;2_6a;  +  5  =  0.  -22.  x2-12a;  =  64. 

19.  a;2_8a._|_7  =  o.  23.  18ar^  +  6a;  =  4. 

20.  2ar^-5a;  =  42.  24.  a.'2-a;-72  =  0. 

21.  l^-\-2x  =  S2.  V26.  4ar'-12a;  =  27,    \ 

▲CAD.    ALG.  —  19 


290  QUADRATIC   EQUATIONS 


26, 
27. 

a^_30  =  13ic. 

38. 

1+x     x-1      4 
a;-3      a;-2      5 

ar^-12a;  =  28. 

39. 

.T         a;  —  5      3 

28. 

x-5         x         2 

29. 
30. 

ar^  +  8  aj  =  84. 

40. 

aj  +  7      a;  +  12      ^ 

31. 

,.1-1=0. 

41. 

a;  +  4o     a;-f3 
a;-2         ~x-3 

32. 

0^2     x_35 
9      3      4* 

42. 

4a;       iK  +  3_^ 

»  — 1              X 

33. 

X           x-2 

43. 

X         1      x  +  2 

#- 

9(a;-l)         6 

x+2      2        2x 

^4. 

4              1 

44. 
45. 

5x    +^  +  6_3^ 
x+7     x+3 

x-j-2     x  +  5     . 
x-7     x-5~    ' 

35. 

a^-2x-\-l      4 

a^     2x     ,.Q 
---=28. 

36. 

^       1^-4. 
2a;  +  l     a;-3 

46. 

a;-3      a;  +  2     23 
a;  +  4  '  a;  -  2     10 

37. 

3  a;  -  1     a;  +  1 
ic  +  2       aj-2 

47. 

2a^4-l      5     a;-8 
l-2a;     7         2 

Literal  Equations 

296.    1.    Solvetheequation  a;2_a^_^^_j_jL  =  o. 

6         a 


Solution 


x2_«x-^a;  +  l=0. 


Factoring,  (x --)lx --\  =  0. 

Therefore,  x  =  ^  or  -. 

6        a 


QUADRATIC  EQUATIONS  291 

Solve  by  any  of  the  preceding  methods : 

2.    x^  —  ax  =  ab  —  bx.  6.    5x  —  2ax  =  a^  —  10a. 

•<  3.   a^ -\- ax  =  ac -{- ex.  "^7.    aP -\-3bx  =  5cx-\-15bc. 

4.  x^  =  (m  —  n)x-^  mn.  8.    2  abx  —  x^  =  14:  ab  —  T  x. 

5.  x^-3bx  =  2ax-6ab.  9.    6x-  +  3aa;  =  2  6a; +  a&. 

10.  acxP  —  box  —  bd  -\-  adx  =  0. 

11.  x^-{-4:mx-\-3nx-\-V2inn  =  0. 

12.  x^  —  2ax  =  a\  ».„_     x  .  a     5 

^^b.    -  +  -  =  -. 

13.  a.-2  +  4&x  =  6l  a      a;      z 

-^14.    a:2  =  4aa;-2a2.  27.    ?L^^l^  =  ^. 

4  3       3 

15.    a^  — aa;  +  a^  =  0. 

2a;  3a    _o 

■>16.    a^  =  6a.  +  il  28.   — -_— __^. 

17.  a^  +  px+?  =  0.  „  2a 

18.  a:2-2a;  +  a  =  0.  a;-a  a; 

19.  4  aa;  —  ar^  =  3  al  i/ 3^         1      ^  i      ^^ 


.  tta;-|-4  16 

20.    ar  —  a  =  1  H- aa;. 

x'21.    aP-h^^a'-bx.  '31.    ar^ _^ «  .^ ^ «_±J^. 

0  0 

^22.   2162_46a;  =  ar*. 

_       „  ,  o      (2a'^  +  l)a; 
^23.    5ax  +  6a'  =  6a^.  ^^'   ^+^  =  - ^— ^• 

^Q     ^      2a;^4(a6-l) 
25.    ar  +  &2^4(a2  +  6a;).  aft  a6 

34.  a;--2(a  — 5)a;  =  4a6. 

35.  x-  +  2(a  +  8)a;  =  — 32a. 

36.  x^  -{-x  +  bx-^b  =  ax  +  a. 

37.  2ax  — a  +  2  6a;  — 6  =  2ar^  — a;. 

38.  a.-2  +  4(a-l)a:  =  8a-4a2. 

39.  a(a;  —  2  a  +  6)  +  a(x  -^a—b)  =  oiP  —  (a  —  by. 


292 


ACADEMIC  ALGEBRA 

40. 

2a+x     a—2x_S 
2a-x     a-{-2x~3 

41. 

1              1          3  +  ar' 
a  —  X      a  -{-  X      d^  —  x^ 

42. 

X  -{-  a     x  -^  b      a  —  b 

43 

x'  4-1      a  -\-b          c 

X             c         a  -{-b 

44. 

2  x  —  a      o         4  a 
b                  2x-b 

45. 

bx      1   ^      a(x-^2b) 
a  —  X  '   ^          a  -\-  b 

46. 

Va  -\-  X  —  Va  —  x  =  V2 

47. 

Va^  —  a  -\-  -Vb  —  X  =Vb 

48.  Va^  -  62  =  Vic  +  ft  Va  +  &. 

49.  V«  —  i»  +  V&  —  £c  =  Va  +  6  —  2  ic. 
50.    Solve  and  verify  Va?  +  1  4-  Va;  —  2  —  V2  a;  —  5  =  0. 


Solution 


VxTl  +  Vx-2  -  \/2a:-  5  =  0. 


Transposing,  Vx  +  1  +  Vx  —  2  =  \/2  a;  —  6. 


Squaring,  a;  +  1  +  2  Vx^^  —  a*.  —  2  +  a;  —  2  =  2a;  —  5. 


Simplifying,  Vx'-^  —  a;  —  2  =  -  2. 

Squaring,  x^  -x-2  =  4. 

Solving,  a;  =  -  2  or  3.  ' 

Verification.  —  Substituting  —  2  for  x  in  the  given  equation, 

that  is,  V^T  +  2  V^^  -  3\/^n^  =  0. 

Therefore,  —  2  is  a  root  of  the  given  equation. 
Substituting  3  for  x  in  the  given  equation, 

Vi  +  Vi  -  vT  =  0, 

which  is  not  true  according  to  the  convention  adopted  in  the  discussion  in 
§  284.     Hence,  3  is  not  to  be  regarded  as  a  root  of  the  given  equation. 


QUADRATIC   EQUATIONS 


293 


Note.  — In  the  previous  verification,  when  only  positive  square  roots  are 
taken,  the  second  value  obtained  for  x  does  not  satisfy  the  given  equation, 
yet  this  indicates  no  error  in  the  process  of  rationalizing,  for  the  equation 
can  be  verified  by  admitting  negative  square  roots.  But,  as  explained 
in  §  283,  it  is  more  convenient  to  regard  3  as  the  root  of  the  equation 
Vx  +  1  —  Vx  —  2  —  \/2x  —  5  =  0,  which  has  for  its  first  member  one  of  the 
rationalizing  factors  of  the  given  equation. 

Solve  and  verify : 


51.  8Va;  — 8a;  =  f. 

52.  3a;-f  Vic  =  5V4 


53.  X  — l+Vx*  +  o  =  0. 

54.  x  —  o—  -^x  —  3  =  0. 


55.    V4  X  +  17  4-  ViC  4-  1  —  'i  =  0. 


*-  56.    Vx  —  1  +  V2  x  —  1  —  V5  a;  =  0. 


57.    V2ic  — 7  —-V^x-if-Vx  —  l  =0. 


V  58.    Va;  +  3  +  V4  x*  +  1  —  VIO  a;  -I-  4  =  0. 

0. 


59.    V6  -f  a;  +  Va;  —  VlO  —  4  a; 
,  60.    V4a;-3  -  V2a;4-2  =  Va;-6. 


61.    V2  a;  +  3  —  Va;  H-  1  =  Vo  a;  —  14. 


62.    V3a;  — 5  4- Va;  — y=V4a;  — 4. 


63.    Var  4  a^  —  Va;  —  2  a^  =  V2  a;  —  5  a^. 


Problems 

297.    1 .    The  sum  of  two  numbers  is  8,  and  their  product  is  15. 
Find  the  numbers. 

Solution 


Let 
Then, 

Since  their  product  is  15, 
Solving, 
and 


X  =  one  number. 
8  —  a;  =  the  other. 
8x-x2=15. 

a;  =  3  or  5, 
8  -  05  =  5  or  3. 


Therefore,  the  numbers  are  3  and  5. 


294 


ACADEMIC  ALGEBRA 


2.  A  party  hired  a  coach  for  ^12.  In  consequence  of  the 
failure  of  three  of  them  to  pay,  each  of  the  others  had  to  pay 
20  cents  more.     How  many  persons  were  in  the  party  ? 

Solution 


Let 

X  =  the  number  of  persons. 

Then, 

JB  —  3  =  the  number  who  paid, 

12 
=  the  number  of  dollars  each  paid, 

d 

12 

—  =  the  number  of  dollars  each  should  have  paid. 

Therefore, 

12        1      12 

x-3      5       X                   ' 

Solving, 

X  =  15  or  -  12. 

and 


The  second  value  of  x  is  evidently  inadmissible.  Hence,  the  number  of 
persons  in  the  party  was  15. 

3.  A  cistern  can  be  filled  by  two  pipes  in  24  minutes.  If  it 
takes  the  smaller  pipe  20  minutes  longer  to  fill  the  cistern  than 
the  larger  pipe,  in  what  time  can  the  cistern  be  filled  by  each 
pipe  ? 


Let 
Then, 

Since 


and 
and 


Solution 

X  =  the  number  of  minutes  required  by  the  larger  pipe. 
a;  +  20  =  the  number  of  minutes  required  by  the  smaller. 

-  =  the  part  which  the  larger  pipe  fills  in  one  minute, 
the  part  which  the  smaller  pipe  fills  in  one  minute, 


Solving, 


ac  +  20 

^  =  the  part  which  both  pipes  fill  in  one  minute, 

X     a;  +  20      24 

x  =  40  or  -  12. 


Hence,  the  larger  pipe  can  fill  the  cistern  in  40  minutes,  and  the  smaller 
in  60  minutes. 

4.  Divide  20  into  two  parts  whose  product  is  96. 

5.  Divide  14  into  two  parts  whose  product  is  45. 

6.  A  man  purchased  a  flock  of  sheep  for  $75.  If  he  had 
paid  the  same  sum  for  a  flock  containing  3  more,  they  would 
have  cost  $  1.25  less  per  head.     How  many  did  he  purchase  ? 


QUADRATIC   EQUATIONS  295 

7.  A  rectangular  garden  is  12  rods  longer  than  it  is  wide, 
and  contains  1  acre.     What  are  its  dimensions  ? 

8.  If  each  side  of  a  square  field  were  lengthened  4  rods,  the 
area  would  be  increased  136  square  rods  more  than  \  of  it.  What 
are  the  dimensions  of  the  field  ? 

9.  A  rectangular  lot  is  8  rods  longer  than  it  is  wide.  What 
are  its  dimensions,  if  it  contains  1^  acres  ? 

10.  In  a  column  of  600  soldiers  each  file  contained  3  men  more 
than  9  times  as  many  men  as  each  rank.  How  broad  and  how 
deep  was  the  column  ? 

11.  A  party  had  a  dinner  that  cost  ^60.  If  there  had  been 
5  persons  more,  the  share  of  each  would  have  been  f  1  less. 
How  many  persons  were  there  in  the  party  ? 

12.  A  man  worked  a  certain  number  of  days  for  $30.  If  he 
had  received  $  1  per  day  less  than  he  did,  he  would  have  been 
obliged  to  work  5  days  longer  to  earn  the  same  sum.  How  many 
days  did  he  work  ? 

13.  Find  two  consecutive  numbers  the  sum  of  whose  squares 
is  61. 

14.  Find  two  consecutive  numbers  the  sum  of  whose  recipro- 
cals is  7^^. 

15.  A  picture  that  was  8  inches  by  12  inches  was  placed  in  a 
frame  of  uniform  width.  If  the  area  of  the  frame  was  equal  to 
that  of  the  picture,  what  was  the  width  of  the  frame  ? 

16.  A  merchant  purchased  a  quantity  of  flour  for  $  100,  and 
retailed  it  at  a  gain  of  f  1  per  barrel.  After  he  had  sold  $  100 
worth  of  it,  he  had  5  barrels  of  it  left.  How  many  barrels  did 
he  buy,  and  at  what  price  ? 

17.  A  merchant  sold  a  hunting  coat  for  f  11,  and  gained  a 
per  cent  equal  to  the  number  of  dollars  the  coat  cost  him.  What 
was  his  per  cent  of  gain  ?  ' 

18.  A  railway  train  traveled  5  miles  an  hour  slower  than 
usual  and  was  one  hour  late  in  making  a  run  of  280  miles.  How 
many  miles  per  hour  did  it  travel  ? 


296  ACADEMIC  ALGEBRA 

19.  A  rectangular  park  56  rods  long  and  16  rods  wide  was 
surrounded  by  a  street  of  uniform  width,  containing  4  acres. 
What  was  the  width  of  the  street  ? 

20.  A  boatman  rowed  8  miles  up  a  stream  and  back  in  3  hours. 
If  the  velocity  of  the  current  was  2  miles  an  hour,  what  was  his 
rate  of  rowing  in  still  water? 

21.  A  man  who  owned  a  lot  56  rods  long  and  28  rods  wide 
constructed  a  road  around  it,  thereby  decreasing  the  area  of  the 
lot  2  acres.     What  was  the  width  of  the  road  ? 

22.  A  man  bought  two  lots  of  cloth  and  paid  96  shillings  for 
each.  There  were  20  yards  in  all,  and  the  number  of  shillings 
per  yard  paid  for  each  was  the  same  as  the  number  of  yards  of 
the  other.     How  many  yards  of  each  did  he  buy  ? 

23.  Find  the  price  of  eggs,  when  2  less  for  30  cents  raises  the 
price  2  cents  per  dozen. 

24.  A  and  B  started  at  the  same  time  and  traveled  toward  a 
place  75  miles  distant.  A  traveled  one  mile  an  hour  faster  than 
B  and  reached  the  place  2^  hours  before  B.  At  what  rate  did 
each  travel  ? 

25.  A  person  drew  a  quantity  of  wine  from  a  cask  filled  with 
81  gallons  of  pure  wine,  and  replaced  it  with  water.  He  then 
drew  from  the  mixture  as  many  gallons  as  he  drew  before  of  pure 
wine,  when  it  was  found  that  the  cask  contained  only  64  gallons 
of  pure  wine.     How  many  gallons  did  he  draw  each  time  ? 

26.  The  circumference  of  the  fore  wheel  of  a  coach  is  5  feet 
less  than  that  of  the  hind  wheel.  If  the  fore  wheel  makes  150 
more  revolutions  than  the  hind  wheel  in  going  a  mile,  what  is 
the  circumference  of  each  wheel  ? 

27.  Two  pipes  running  together  can  fill  a  cistern  in  2|  hours. 
The  larger  pipe  can  fill  the  cistern  in  2  hours  less  time  than  the 
smaller.  How  many  hours  will  it  take  each  pipe  alone  to  fill 
the  cistern  ? 

28.  It  took  a  number  of  men  as  many  days  to  dig  a  ditch  as 
there  were  men.  If  there  had  been  6  more  men,  they  would  have 
done  the  work  in  8  days.     How  many  men  were  there  ? 


QUADRATIC  EQUATIONS  297 

EQUATIONS   IN   THE   QUADRATIC   FORM 

298.  An  equation  that  contains  but  two  powers  of  an  unknown 
number  or  expression,  the  exponent  of  one  power  being  twice 
that  of  the  other,  is  in  the  Quadratic  Form. 

Equations  in  the  quadratic  form  can  be  reduced  to  the  general 
form  aa^  +  fex"  -|-  c  =  0,  in  which  n  represents  any  number. 

Examples 
1.   Given  a;^  +  6  ic^  —  40  =  0,  to  find  the  values  of  x. 

Solution 
a^  +  6  ic2  _  40  =  0. 
Factoring,  (x^  -  4)  (a;2  +  10)  =  0. 

.-.  a;2  -  4  =  0  or  x'^-\-\Q  =  0, 


and  x  =  ±1  or   ±V-  10. 

2.    Given  ic^  —  a;*  =  6,  to  find  the  values  of  x. 
First  Solution 
x^  -x^  =  6. 
Completing  the  square,   x^  —  x*  -{■  (i)^  =  ■^. 
Extracting  the  square  root,  x*  —  \=±^. 


.-.  x*  =  3  or  -  2. 

Raising  to  the  fourth 

power,                X  =  81    or  16. 

Second  Solution 

:.i-=.i  =  6. 

Let  X*  =  p, 

then. 

x^  =  p2,  and  p2  _  p  =  6. 
.-.  i)2  _  p  _  6  _  0. 

Factoring, 

(i>-3)(p  +  2)  =  0. 

.-.  p  =  3  or  —  2 ; 

that  is, 

aji  =  3  or  -  2. 

Whence, 

X  =  81  or  16. 

298  ACADEMIC  ALGEBRA 

3.  Solve  the  equation  a;  —  4  i»^  -1-  3  a;^  =  0. 

Solution 

Let  x^  =  j),  then,            x^  =  p^,  and  x  =  p^. 

Then,  p^  -  4:p^  +  3p  z=0. 

Factoring,  p(p'^  —  4:  p  +  S)  =  0. 

Whence,  P  =  0, 

or  p^-4:p-\-S  =  0. 

Factoring,  .                (i?  -  1)  (P  -  3)  =  0. 

Whence,  p  =  I  or  p  =  S. 

That  is,  x^  =  0,  1,  or  3. 

.-.  x  =  0,  1,  or  27. 

4.  Given  x^-7x-{- ^x^ -  7  a;  + 18  =  24,  to  find  the  value  of  x. 

Solution 


x^-7x  +  Vx-^-lx  +  18z=  24. 

(1) 

Adding  18, 

x2  -  7  a;  +  18  +  ■\/a;2  -  7  x  +  18  =  42. 

(2) 

Put  p  for  (a;2 

-  7  a;  +  18)^  and  p^  for  (x'^  -  7  x  +  18). 

(3) 

Then, 

i)2+p_42  =  0. 

(4) 

Solving, 

j9  =  6  or  -  7. 

(5) 

That  is. 

Va;2  -  7  X  +  18  =  6, 

(6) 

or 

Va;2  -  7  X  +  18  =  -  7. 

(7) 

Squaring  (6), 

a:2  -  7  X  +  18  =  36. 

Solving, 

X  =  9  or  -  2. 

Squaring  (7), 

a;2  -  7  a;  +  18  =  49. 

Solving, 

xr=|±  iVl73. 

Hence, 

x  =  9,   -2,  or  |±iVl73. 

5.    Solve  the  equation  a;**  —  9  ar'  +  8  =  0. 

Solution 

(1) 
(2) 
(3) 
(4) 


aj6-9aj3  +  8  =  0. 

Factoring, 

(aj8-l)(x3-8)=0. 

Therefore, 

aj8  -  1  =  0, 

or 

x8  -  8  =  0. 

QUADRATTC  EQUATIONS  299 

If  the  values  of  x  are  found  by  transposing  the  known  terms  in  (3)  and 
(4)  and  extracting  the  cube  root  of  each  member,,  only  one  vahie  of  x  will 
be  obtained  from  each  equation.  But  if  the  equations  are  factored,  three 
values  of  x  are  obtained. 

Factoring,  (x  -  1)  (ic2  +  x  +  1)  =  0,  (5) 

and  (x-2)(x2  +  2x  +  4)  =  0.  (6) 

Writing  each  factor  equal  to  zero,  and  solving : 

From  Eq.  (5),    x  =  1,    -\  +  \-/^^,  -\-  ^V^^.  (7) 

From  Eq.  (6),    x  =  2,   -  1  +  V^   -  1  -  yT^.  (8) 

Since  the  values  of  x  in  (7)  are  obtained  by  factoring  x^  —  1  =  0,  they 
may  be  regarded  as  the  three  cube  roots  of  the  number  1.  Also,  the  values 
of  X  in  (8)  may  be  regarded  as  the  three  cube  roots  of  the  number  8  (§  225). 

6.    Solve  the  equation  a;^4-4a:^  —  8a;  +  3  =  0. 

Solution 
Extracting  the  square  root  of  the  first  member  as  far  as  possible, 

x*  +  4x3-8x  +  3  I  x2  +  2  X  -  2 

X* 


2  x2  +  2  X  I    4  x8 
2  x2  +  4  X 


4x8 

'  +  4x2 

■2| 

-4x2- 
-4x2- 

-8x  +  3 
-  8  X  +  4 

-  1 

Since  the  first  member  lacks  1  of  being  a  perfect  square,  the  square  may- 
be completed  by  adding  1  to  each  member,  giving  the  following  equation  : 

x4  +  4x8-8x  +  4  =  l. 
Extracting  the  square  root,       x2  +  2x  —  2  =  ±1. 

.-.  x2  +  2  X  -  3  =  0,  and  x2  +  2  X  -  1  =  0. 
Solving,  X  =  1,   -  3,   -  1  ±  V2. 

Solve  the  following  equations  : 

7.  ic*-13x-2-h36  =  0.  11.  5  a;*  4- 6  3^-11  =  0. 

8.  «^-25ar^  +  144  =  0.  12.  2  x^ -d>  x" -9{^  =  0. 

9.  iK^-18a^4-32  =  0.  13.  a;^  _  5  .r^  +  6=  0. 
^10.    3  a;*  +  5  x2  -  8  =  0.  ^^  14.  oj^  +  3  a;^  -  28  =  0. 


300  QUADRATIC   EQUATIONS 

"15.    a;*-3£C*  =  -2.  25.    ^-3aj^H-2iB^  =  0. 

^16.    x^  —  x^=^^.  26.    5  a;  =  icVaJ  +  6 V^. 

Jl7.    a;  +  2Vic  =  3.  27.    3aj=  a;^^  +  2^^. 

'  18.    x^- 2x^  =  3.  '  28.    x-^  -  3  -  4  x^  =  0. 

^19.    a.-3  4-8a;^-9  =  0.  29.    a;"^  -  6  a;^  =  1. 

20.    a.t  +  ,,f_2  =  0.  30.    x-^  +  x' =  2 x'K 

'21.    ^^  +  3V^  =  30.  31.    a;  +  2a;^  =  3a;t 

''22.    aa^''  +  fex"  +  c  =  0.  32.    2  a;  +  V«  =  15  xVx. 

^23.    a;^  — 4  a;  — 5a;^  =  0.  33.    V^  + 5  +  6  a"^  =  0. 

•*24.    a;^-a^- 12x^  =  0.  *34.    a;*  =  8  a;  +  7  a^ V^. 

35.    (a;-3)2  4-2(a;-3)  =  3. 
-36.    (ar^  +  l)2  +  4(a^  +  l)  =  45. 
37.    (a;2_4)2-3(a^-4)  =  10. 
.38.    (a^-2a;)2-2(a^-2a;)  =  3. 
39.    (a^-xy-{x'-x)-132  =  0. 


40.    x—5-]-  2Vx  —  5  =  8. 


41.    a^-3a;  +  6+2Vaj'-3a;  +  6  =  24. 


42.    a^-5a;  +  2Va^-5a;-2  =  10. 


43.    a^  —  a;—Va^— a; +  4  — 8=0. 


44.    a^-5x  +  5Va:'-5a;  +  l  =  49. 


^45.  a!4-10  =  2Vaj  +  10  +  5.  49.  «  +  2Va;  +  3  =  21. 

46.  a;  -  3  =  21  -  4 Va;  -  3.  50.  2x-  3 V2 x-\-5  =  -5. 

^47.  2  a;  -  6 V2¥^"l  =  8.  51.  a^  +  «V^-72  =  0. 

48.  aj  =  11  -  3  Va;  +  7.  52.  a;"^  -  5  a;"^  +  4  =  0. 


17 
16* 


QUADRATIC  EQUATIONS  301 

58.  (a;-a)^-3a^(a;-a)*  +  2a^  =  0. 

^  59.  Find  the  three  cube  roots  of  —  1. 

"^  60.  Find  the  three  cube  roots  of  —  8. 

61.  Find  the  four  fourth  roots  of  1. 

Solve  the  following  equations : 

62.  a;«-28a^  +  27  =  0.  65.    a^  +  2a^- a;  =  30. 

Q  66.    0^  —  40^ +  8 a;  =-3. 

63.  Q^--^  =  7. 

x"  67.    aj*-2ar^  +  iB  =  132. 

64.  a;^-16  =  0.  68.   a;^  -  6a^  +  27a;  =  10. 

69.  a;*  +  2iB^  +  5ar*  +  4a;-60  =  0. 

70.  a;^  +  6a«4-7ar'-6a;-8  =  0. 

71.  a;<-6.a^  +  15a.-2-18a;  +  8  =  0. 

72.  iB*-10a^  +  35«^-50a;  +  24  =  0. 

73.  16a^-8ar^-31ic2^83.^15^0. 

74.  4aj^-4ar^-7a^  +  4ic4-3  =  0. 

75     _^ x-\-l^  7, 

'    cB-fl         a^        12 

Suggestion.  —  Since  the  second  term  is  the  reciprocal  of  the  first,  put  p 

for  the  first  term  and  —  -  for  the  second. 
P 

Then,  p  _  i  =  JL. 

^     p      12 

76.    ^±^  +  _A_  =  2  78.   £+2      2(^+4)^51. 

2  a:^^^       •  '  a^  +  4        a;  +  2  5 

77     ^'  +  1  I       4      ^5  fl^  +  l      4(a;-l)^21 

*       4      "^ar^  +  l      2*  '    a;-l        ar^  4- 1        5 


302  ACADEMIC  ALGEBRA 


82.    ot^-3x-i-- -  =  1.      85.    x^-2x-{- 

ocr  —  Sx-{-2 

86.    .        ^         .+  2 


l  +  a;  +  aj2      Vl+aj  +  ic^ 


SIMULTANEOUS  EQUATIONS  INVOLVING 
QUADRATICS 

299.  An  equation  whose  terms  are  homogeneous  with  respect 
to  the  unknown  numbers  is  called  a  Homogeneous  Equation. 

x^  —  xy-\-y^  =  6,  x^-\-y^  =  12,  and  ax-j-y  =  10  are  homogeneous  equations. 

300.  An  equation  that  is  not  affected  by  interchanging  the 
unknown  numbers  involved  in  it  is  called  a  Symmetrical  Equation. 

2  x^  +  xy  +  2  y^  =  32,  x^  +  y^  =  2S  are  symmetrical  equations. 

301.  Many  simultaneous  equations  involving  quadratics  may 
be  solved  by  the  rules  for  quadratics,  if  they  belong  to  one  of  the 
following  classes : 

1.  When  one  is  simple  and  the  other  quadratic. 

2.  When  the  unknown  numbers  are  involved  in  a  similar  manner 
in  each  equation. 

3.  When  each  equation  is  homogeneo^is  and  quadratic. 

I.    Simple  4ud  quadratic. 


1.   Solve  the  equations   , 

Solution 

x-Vy  =1.  (1) 

3x2  +  ?/2  =  43.  (2; 

From  (1),  y  =  7  -x.  (3 

Substituting  in  (2),  3x2  +  (7  -  ^y  ^  43.  ^4^ 


a;  =  3  or  i. 

(6; 

y  =  4. 

(6) 

y  =  ¥. 

(7) 

(  when  X  =  3,  y  =  4, 
when  X  =  ^,  y  =  ^^' 

QUADRATIC  EQUATIONS  303 

Solving, 

Substituting  3  for  x  in  (3), 

Substituting  ^  for  x  in  (3), 

That  is,  X  and  y  each  have  two  values 

Equations  of  this  class  may  be  solved  by  finding  the  value  of 
one  unknG\»rn  number  in  terms  of  the  other  in  one  equation,  and 
then  substituting  it  in  the  other. 

Solve  the  following  equations : 

rar^  +  2/'  =  20,  p  =  6-.y, 

*  [x  =  2y.  '  {ar^  +  ^  =  72. 

(10x  +  y  =  3xy,  ^     (xy{x-2y)  =  10, 

'    [y-x  =  2.  '   la^  =  10. 

^     (x'-\-xy  =  12,  ^     r  3  35(2/  +  !)  =  12, 

*  [x-y  =  2.  '  l3a;  =  23,. 

II.   Unknown  numbers  similarly  involved, 

{X   \    y  —  7 
xy  =  10. 

Solution 

x  +  y  =  7.  (1) 

xy  =  10.  (2) 

Squaring  (1),                       x"^ -\- 2  xy -\- y^  =  49.  (3) 

Multiplying  (2)  by  4,                          4  x?/  =  40.  (4) 

Subtracting  (4)  from  (3),  x^-2xy  +  y^  =  9.  (5) 

Extracting  the  square  root,              x  —  y  =  ±S.  (6) 
From  (l)  +  (6),                                          a;  =  5  or  2. 
From  (l)-(6),           ^                               y  =  2  or  6. 

Such  equations  may  be  solved  by  the  method  illustrated  in 
example  1,  but  the  method  given  above  of  finding  the  value  of 


304  ACADEMIC  ALGEBRA 

x  —  y,  so  that  it  may  be  combined  with  the  vahie  of  x-\-y  to  dis- 
cover the  values  of  x  and  y,  is  preferable. 


9.    Solve  the  equations 

(x^^f==25, 
[x-\-y  =  7. 

Solution 

a;2  + 2/2  =  25. 

0) 

x  +  y  =  7. 

(2;) 

Squaring  (2), 

a;2  ^2xy-\-y^  =  49. 

(3) 

Subtracting  (1)  from  (3), 

2xy  =  24, 

(4) 

Subtracting  (4)  from  (1), 

x:2-2xy  +  y^  =  l. 

(6) 

Extracting  the  square  root, 

x-y=±l. 

(6) 

From  (2) +  (6), 

a;  =  4  or  3. 

From  (2)  -  (6) 

y  =  3  or  4. 

fx*  -\-  y*  =  97 
a;  +  2/  =  1. 

Solution 

x*-\-y*  =  97.  (1) 

x-\-y  =  l.  (2) 

4tli  power  of  (2), 

X^  +  ix^y  +  6  X2y2  ^.  4  a;y3  _,_  y4  =  1,  (3) 

Subtracting  (1) ,           4  x^y  +  ^  x^y'^ -{■  ^  xy^  =  -  96  (4) 

Dividing  by  2,              2x^y  +  S  x^y^  +  2xy^  =  -  48.  (5) 

2xy  X  square  of  (2) ,   2  x^y  +  4  x2y2  +  2  xy^  =  2  xy.  (6) 

Subtracting  (5)  from  (6),            x2y2  _  2  xy  =  48.  (7) 

Solving  for  xy,                                           xy  =—6  or  S.  (8) 
Equations  (2)  and  (8)  give  two  pairs  of  simultaneous  equations, 


[  xy  =  -  6  [xy  =  S 


Solving  these  by  previous  methods, 


x  =  3,   or   -2,   or  ^  +  ^V-Tsi,   or  ^  -  ^V-TsT. 
y=-2,  or  3,   or  ^-l^/:^3I,   or   1  +  ^V^^^. 


QUADRATIC  EQUATIONS  305 

Solve  the  following  equations : 

11.     i  14.    < 

[xy  =  7.  [  X -{- y  -\- xy  =  11. 

.  +  ,  =  8,  15.   P^  +  ^^=/'' 

13.    J^  +  2/  =  9,  ^g^    fa^^  +  a^2/^  +  2/*  =  21, 


[a^  +  y'  = 


f  =  243.  I  ar^  +  a;?/  +  2/^  =  7 

III.   Homogeneous  quadratics. 

x-  —  xy-{-y^  =  21, 


17.    Solve  the  equations   ,    «      ^ 

'  y^  —  2  a:^  =  —  lo. 


(1) 

(2) 
(3) 
(4) 
(5) 

(6) 

(7) 

(8) 
2  I?  -  1      o^  -  V  +  1  ^ 

Clearing,  etc.,  5  r2  -  19 1)  + 12  =  0.  (9) 

Factoring,  (v  -  3)  (5  v  -  4)  =  0.  (10) 

.'.  v  =  S  or  f  (11) 

Substituting  3  for  v  in  (7)  or  in  (6),     y  =  ±Vs   | 

and  since  x  =  vy,  x  —  ±3  V3  I 

Substituting  f  f or  v  in  (7)  or  in  (6) 

and  since  x  =  vy^  x=± 

a;=+3\/3,  or  —  3>/3,  or  4- 4,  or  —4, 


Solution 

a;2 

-  a;y  +  y-  =  21. 
y^-2xy=-  15. 

Assume 

X  =  vy. 

Substituting  in  (1), 

^?v 

-  vy2  +  y2  :=  21. 

Substituting  in  (2), 

y2_2^?y2=-15. 

Solving  (4)  for  y\ 

1/2-         21 

^          d2  _  ^  +  1 

Solving  (5)  for  y'^. 

V2-         1^       . 

2t;-  1 

Comparing  the  values  of  y^^ 

15                 21 

»,     y=±5| 
x=±4  J 


Hence,  , 

y  =  +  V3,  or  —  V  3,  or  +  5,  or  —  6. 


ACAD.    ALG. — 20 


306  ACADEMIC  ALGEBRA 

Solve  the  following  equations : 


^18. 


a!2/  +  33(''  =  20,  f  of  -  xy  -  y^  =  20, 

21 


■■( 


x^-^xy  =  12,  ^^^     {^-xy  +  f  =  21, 

a;?/ +  2  2/2  =  5.  I  0^  + "2  2/2  =  27. 

«2  +  2  2/'  =  44,  r  2  a^  -  3  0^2/  +  2  2/'  =  100, 


>>  23. 

xy-y^  =  S.  [x'-y^  =  75. 


302.  Many  simultaneous  equations  that  belong  to  one  or  more 
of  the  preceding  classes,  and  many  that  belong  to  none  of  them, 
may  be  readily  solved  by  special  devices. 

(x^  +  xy  =  12, 

24.  Solve  the  equations    -i 

U2/  +  2/^  =  4. 

Solution 

x'^  +  xy  =  12.  (1) 

xy-}-y^  =  ^.  '                  (2) 

Adding,                       x^ -^  2  xy -\- y^  =  16.  (3) 

.-.  x  +  y  =  -{-  4:  or  —  4.  (4) 

Subtracting  (2)  from  (1),      x^  -y'^  =  S.  (6) 

Dividing  (5)  by  (4),                 x-y=+2oT-2.  (6) 

Combining  (4)  and  (6),  a;  =  3  or  -  3  ;  y  ==  1  or  —  1. 

(x*-\-a^y^  +  y*  =  AS, 

25.  Solve  the  equations    i    „ 

^  [x^-xy-\-y'  =  12. 

Solution 

JC4   +   a;2j/0  ^yi^  48,  (1) 

x2  -  xy  +  ?/2  ^  12.  (2) 

Dividing  (1)  by  (2),     x^  +  xy  +  y^  =  4.  (3) 

From  (3) -(2),  xy  =  -4:,  (4) 

By  adding  (4)  to  (3),  and  subtracting  (4)  from  (2),  the  values  of  (x  +  yY 
and  {x  —  yy  may  be  found  and  the  solution  readily  completed. 


QUADRATIC  EQUATIONS  307 

f  a^  -  2/^  =  26, 
26.    Solve  the  equations    \ 

Solution 

y?-y^  =  26.  (1) 

x-y  =  2.  ^                                         (2) 

Dividing  (1)  by  (2),     x'^  +  xy  ^-  y'^  =  13.  (3) 

Squaring  (2),               x'^  -  2  xy -\- y^  =  i.  (4) 

Subtracting,  dividing  by  3,           xy  =  3.  (5) 

By  adding  (5)  to  (3),  the  value  of  (x  +  y)'^  may  be  obtained  and  the 

solution  completed  as  in  previous  examples. 


27.    Solve  the  equations    i 

\    x  —  y  =  2. 


Solution 

X*  +  y*  =  82.  (1) 

x-y  =  2.  (2) 

Assume  x  =  u -\-  v,  (3) 

and  y  =  ti-v.  (4) 

Substituting  these  values  in  (1), 

M*  +  iuH  +  Q  uH^  +  4  wy3  +  «* 

+  M*  -  4  W8v  +  6  WV  -  4  WV3  +  |;4  =  82,  (5) 

and  in  (2),                                         2  v  =  2.  (6) 

Dividing  (5)  by  2,                  m*  +  6  mV  +  t;*  =  41.  (7) 

Dividing  (6)  by  2,                                         v  =  l.  (8) 

Substituting  1  for  v'in  (7)  and  solving,  w  =  ±  2  or  ±  V-  10.  (9) 
Hence,  from  (3)  and  (4),       x  =  3  or  -  1  or  1  ±  V-HO, 

and  y  =  lor-3or-l±  V-  10. 

f  or'  4-  2/^  +  a;  4-  y  =  14, 
28.    Solve  the  equations    \ 

[xy  =  S. 

Solution 

x^  +  y^  +  x-\-y-U. 
xy  =  S. 
Adding  twice  the  second  equation  to  the  first, 

x2  4  2  xy  +  2/2  4  X  +  2/  =  20. 


308 


ACADEMIC  ALGEBRA 


Completing  the  square,  {x+  yY  +  x -\- y  ■\-  (^)2  =  20|. 

Extracting  the  square  root,  x  +  y-{-^=.±\. 

.'.  x  +  y  =  4:  or  —  6. 

The  sohition  may  be  completed  by  solving  the  equations, 

(x  +  y  =  4:  (x  +  yz=-6 

\  and  i  _ 

I      xy  =  S         [      ^y  =  ^ 

The  student  will  doubtless  discover  many  other  methods  for 
solving  simultaneous  equations.  All  the  preceding  solutions  are 
but  illustrations  of  devices  that  are  important  only  because  they 
are  often  applicable. 


Solve  the  following  simultaneous  equations 
a^  +  2/'  =  53, 


29. 


^  30. 


31. 


32.    ^ 


2/ =  5. 

ra^  +  2/3  =  28, 

\x-\-y  =  4:. 

■  1  +  a;  =  2/, 

l-\-x  =  y, 
l  +  ^  =  t 


,^^^^.^  +  ,^  =  40, 
xy  =  12. 

x(x  +  y)  =  x, 


'■{ 


34. 


35. 


(x{ 


x-y)=-l. 

x^-{-Sxy-y^  =  ^S, 
x-\-2y  =  10. 

+  2/^  =  19, 


^'36.   1^  +  "^^ 
[QC^  —  y^  = 


19. 


_     (x'  +  3xy  =  f  +  2S, 
37.    i 


38. 


40. 


41, 


42. 


43. 


44. 


45. 


46. 


2a^-\-xy-5y^  =  20, 
2x-Sy  =  l. 


y?  .  4a; 


39.      2/ 


+ 


y 


21, 


I  a;  -  2/  =  2. 

r  a;^  —  3^2/  =  48? 
I  a72/  —  2/^  =  12. 

r  a;2  _|.  a^  =  _  6, 

I  a;y  +  2/^  =  15. 

r  4  3^2/  =  96  -  a;22/^ 
1  a;  +  y  =  6. 

r  a;2  -  a;2/  =  8, 
\xy^f  =  \2. 

I  a;2  _  3^2/  =  6, 
W4. 2/2  =  61. 

|a;2_|_a;2/  =  77, 
{xy  —  y"^  =  12. 

2a;-2/  =  2, 
2ar^  +  2/2  =  f. 


QUADRATIC  EQUATIONS 


309 


47. 


48. 


49. 


50. 


51. 


52. 


53. 


54. 


55 


■( 


56. 


57. 


58. 


2xy-if  =  12, 
3xy-\-oa^  =  104. 

a^ -[- xy -\- y^  =  151, 
x'-\-y^  =  106. 

'  x^ -{- xy -\- y^  =  M, 
.  X  —  ^xy  +  ?/  =  6. 

4ic2-2iC2/  +  2r  =  13, 
I  8  a^  +  2^  =  65. 

'  6  a^  +  6  2/2  =  13  icy, 
.  ar^  -  ?/2  ^  20. 

.  x  +  2/  +  3*2/  =  11. 

'3xy-\-2x  +  y  =  25, 
9x  _4:y 

[  y      » 

ar^  +  a^  =  40, 
27  +  22/=^  =  3a^. 

xy^  +  xy  =  24, 
a^  +  a;  =  56. 

a;^  +  2/4  =  82, 
a;  +  y  =  4. 

x*-y'  =  369, 
ar^  -  1/2  ^  9^ 


/2  _  78  — 


8  =  a;  +  y. 


fa^  +  2/' 

I  a;2/  +  a?  +  y  =  39. 


60. 


61. 


62. 


63. 


64. 


a^  _l_  2  3^2/  +  3  2/2  =  43, 
2a^  +  3a;2/  +  42/2  =  62. 

x^  —  xy  -\-2y^  =  46, 
a.-2  4-  a^  +  3  2/2  =  111. 

a^  -  7  a;y  +  12  2/2  =  0, 
xy-{-3y  =  2x-\-21. 

a^-f  =  31, 
xy(y-x)  =  -12. 

a;  +  2/  =  25, 

i  ^x  +  Vy  =  7. 


ra^  +  2/3  =  2252/, 
•   lar^-7/2  =  75. 

I 


66. 


67. 


68. 


69. 


70. 


x'^xy-\-2f  =  ll, 
2x'-\-5y^  =  22. 

x^  +  y-  =  3xy-^5, 
[a^-\-y*  =  2. 

ar'  +  2/^  =  _i2_, 
x-y 

m^y  —  xy^  =  ^' 

'(a?  +  2/)(^  +  y')  =  65, 
{{x-y){^~f)  =  5. 

x2  +  y  =  a:-2/2  +  42, 
L  a^  =  20. 


^^    |a;  +  2/4-2Va;  +  2/  =  24, 
I  a;  —  2/  +  3Va;  — 2/  =  10. 


I  a^  +  2/2  +  6  V^+  2/'  =  55, 


310 


ACADEMIC  ALGEBRA 


y 


2x     ,  a^  —  afy _S 


73. 


74. 


75. 


76. 


77. 


+ 


f-^y    8 


y       x  +  y 

a^  =  2/2  +  16. 

x^  —  xy  =  a^  -\-  6^, 

xy  —  y"^  =  2  ah. 

x  —  2yz=z2{a-{-h), 

xy^2f  =  2h{h-a). 
'a^  +  2/3  =  2a(a2  +  3  62), 
.  x^y  -\-  xy'^  =  2  a  (a^  —  b^). 

a  —  X     b  -\-y  _  ay  4^6^ 
X  y  (^  —  W 

a^  +  2/2  =  2(a2  +  62). 


Problems 

303.  1.  The  sum  of  two  numbers  is  12,  and  their  product  is 
32.     What  are  the  numbers  ? 

2.  The  sum  of  two  numbers  is  17,  and  the  sum  of  their 
squares  is  157.     What  are  the  numbers  ? 

3.  The  difference  of  two  numbers  is  1,  and  the  difference  of 
their  cubes  91.     What  are  the  numbers  ? 

4.  The  sum  of  two  numbers  is  82,  and  the  sum  of  their  square 
roots  is  10.     What  are  the  numbers  ? 

5.  It  takes  52  rods  of  fence  to  inclose  a  rectangular  garden 
containing  1  acre.     How  long  and  how  wide  is  the  garden  ? 

6.  The  sum  of  the  squares  of  the  terms  of  a  fraction  is  89, 
and  the  fraction  is  |-|  larger  than  its  reciprocal.  What  is  the 
fraction  ? 

7.  Find  two  numbers  such  that  their  product  is  8  greater 
than  twice  their  sum  and  48  less  than  the  sum  of  their  squares. 

8.  If  63  is  subtracted  from  a  certain  number  expressed  by 
two  digits,  its  digits  will  be  transposed;  and  if  the  number  is 
multiplied  by  the  sum  of  its  digits,  the  product  will  be  729. 
What  is  the  number  ? 


QUADRATIC  EQUATIONS  311 

9.  A  man  expended  $  6  for  canvas.  Had  it  cost  4  cents  less 
per  yard,  he  would  have  received  5  yards  more.  How  many 
yards  did  he  buy,  and  at  what  price  per  yard  ? 

10.  If  the  difference  of  two  numbers  multiplied  by  the 
greater  is  160,  and  multiplied  by  the  less  is  96,  what  are  the 
numbers  ? 

11.  A  rectangular  flower  garden  containing  54  square  rods 
was  enlarged  to  twice  its  former  size  by  making  an  addition 
of  1\  rods  on  all  sides.  What  were  the  original  dimensions  of 
the  garden? 

12.  If  it  requires  200  rods  of  fence  to  inclose  a  rectangular 
field  of  15  acres,  what  are  its  dimensions  ? 

13.  A  rectangular  field  contains  20  acres.  If  its  length  were 
20  rods  less  and  its  width  8  rods  less,  its  area  would  be  8  acres 
less.     What  are  its  dimensions  ? 

14.  A  man  found  that  he  could  buy  16  more  sheep  than  cows 
for  $  100,  and  that  the  cost  of  3  cows  was  $  15  greater  than  the 
cost  of  4  times  as  many  sheep.     What  was  the  price  of  each  ? 

15.  Eight  persons  contributed  $30  to  pay  for  a  set  of  books. 
One  half  of  the  amount  was  contributed  by  women,  and  the  other 
half  by  men,  each  man  giving  $  2  more  than  each  woman.  What 
did  each  woman  and  what  did  each  man  contribute  ? 

16.  A  man  loaned  $  1000  in  two  unequal  sums  at  such  rates 
that  both  sums  yielded  the  same  annual  interest.  The  larger 
sum  at  the  higher  rate  of  interest  would  have  yielded  $36 
annually,  the  smaller  sum  at  the  lower  rate,  $16  annually. 
What  sums  did  he  invest,  and  at  what  rates  of  interest? 

17.  If  2  is  added  to  the  numerator  and  subtracted  from  the 
denominator  of  a  certain  fraction,  the  result  will  be  the  recipro- 
cal of  the  fraction;  if  3  is  subtracted  from  the  numerator  and 
added  to  the  denominator,  the  result  will  be  ^  of  the  original 
fraction.     What  is  the  fraction  ? 

18.  The  product  of  two  numbers  is  59  greater  than  their  sum, 
and  the  sum  of  their  squares  is  170.     What  are  the  numbers  ? 


312  ACADEMIC  ALGEBRA 

19.  A  purse  contained  $50.50  in  gold  and  silver  coins.  If 
there  were  fifteen  coins,  and  if  each  gold  coin  was  worth  as  many- 
dollars  as  there  were  silver  coins  and  each  silver  coin  was  worth 
as  many  cents  as  there  were  gold  coins,  how  many  coins  of  each 
kind  were  there  ? 

20.  Two  men  working  together  can  complete  a  piece  of  work 
in  6f  days.  If  it  would  take  one  man  3  days  longer  than  the 
other  to  do  the  work  alone,  in  how  many  days  can  each  man  do 
the  work  alone  ? 

21.  The  fore  wheel  of  a  carriage  makes  12  revolutions  more 
than  the  hind  wheel  in  going  240  yards.  If  the  circumference 
of  each  wheel  were  one  yard  more  than  it  is,  the  fore  wheel 
would  make  8  revolutions  more  than  the  hind  wheel  in  going  240 
yards.     What  is  the  circumference  of  each  wheel  ? 

22.  A  sum  of  money  on  interest  for  1  year  at  a  certain  per 
cent  amounted  to  $11130.  If  the  rate  had  been  1%  less  and 
the  principal  $  100  more,  the  amount  would  have  been  the  same. 
Find  the  principal  and  rate. 

23.  A  number  multiplied  by  another  composed  of  the  same 
two  digits,  but  reversed,  gives  a  product  of  4032.  If  the  first 
divided  by  the  second  is  equal  to  1|,  what  are  the  numbers  ? 

24.  The  town  A  is  on  a  lake  and  12  miles  from  B,  which  is 
4  miles  from  the  opposite  shore.  A  man  rows  across  the  lake 
and  walks  to  B  iu  3  hours.  In  returning  he  walks  at  the  same 
rate  as  before,  but  rows  2  miles  an  hour  less  than  before.  If  it 
takes  him  5  hours  to  return,  find  his  rates  of  walking  and  rowing, 

25.  A,  B,  and  C  started  together  to  ride  a  certain  distance.  A 
and  C  rode  the  whole  distance  at  uniform  rates,  A  two  miles  an 
hour  faster  than  C.  B  rode  with  C  for  20  miles,  and  then,  by 
increasing  his  speed  two  miles  an  hour,  reached  his  destination 
40  minutes  earlier  than  C  and  20  minutes  later  than  A.  Find 
the  distance,  and  the  rate  at  which  each  traveled. 

26.  Find  two  numbers  such  that  their  product  is  equal  to  the 
difference  of  their  squares,  and  the  difference  of  their  cubes  is 
equal  to  the  sum  of  their  squares. 


QUADRATIC  EQUATIONS  313 

PROPERTIES   OP   QUADRATICS 

304.    Every  quadratic  equation  may  be  reduced  to  the  form 

as?  -\-hx-\-c  =  0,  (1) 

in  which  a  is  positive  and  h  and  c  are  positive  or  negative. 
Denote  the  roots  by  r^  and  rg.     Then,  §  294,  Ex.  13, 


'2a  2a  ^^ 

In  the  following  discussion  of  the  nature  of  the  roots  of  a  quadratic  equa- 
tion, the  student  should  keep  in  mind  the  distinctions  hetween  rational  and 
irrational,  real  and  imaginary.  For  example,  2  and  Vi  are  rational^  and  real 
also  ;  v^  and  Vb  are  irrational,  but  real ;  >/—  2  and  V  —  5  are  irrational^ 
and  also  imaginary. 

1.  Suppose  that  b^  —  4: acts  positive. 

Then,  V6^  —  4  ac  is  a  positive  real  number  and  —  V6^  —  4  ac  is 
a  negative  real  number.     Hence,  the  roots  are  real  and  unequal. 

If  6^  —  4  ac  is  a  perfect  square,  the  roots  are  rational  j  otherwise 
they  are  irrational. 

2.  Suppose  that  6^  —  4  ac  =  0. 

Then,  V&^  —  4  ac  =  0  and  the  roots  are  real  and  equal. 

3.  Suppose  that  6^  —  4  ac  is  negative. 

Then,  Vft^  —  4  ac  and  —  ^b^  —  4  ac  are  imaginary,  and  conse- 
quently both  roots  are  imaginary. 

Principles.  —  1.  In  any  quadratic  equation  aa?  +  6a5  -f  c  =  0,  if 
6^  —  4  ac  is  positive  J  the  roots  are  real  and  unequal;  if  b^—4:  ac=0, 
the  roots  are  real  and  equal;  if  6^  —  4ac  is  negative,  both  roots  are 
imaginary. 

2.  If  b^  —  4:ac  is  a  perfect  square  or  is  equal  to  zero,  the  roots  are 
rational;  otherwise  they  are  irrational. 

When  the  roots  are  real,  their  signs  are  found  by  comparing  the 
values  of  b  and  c. 

If  c  is  positive,  —  6  is  numerically  greater  than  ±  V^"  —  4  oc, 
whence  both  roots  have  the  sign  of  —  & ;  if  c  is  negative,  —  6  is 
numerically  less  than  ±  V6^  —  4  ac^  whence  r^  is  positive  and  r^,  is 


314  QUADRATIC   EQUATIONS 

negative.     The  root  having  the  sign  opposite  to  that  of  h  is  the 
greater  numerically.     Hence, 

Principle  3.  —  If  c  is  positive,  both  roots  have  the  sign  opposite 
to  that  of  h ;  if  c  is  negative,  the  roots  have  opposite  signs,  and  the 
numerically  greater  root  has  the  sign  opposite  to  that  of  h. 

The  following  are  special  cases : 

1.  If  c  =  0,  (1)  reduces  to  the  form  ax^  +  6a;  =  0,  which  has  two  roots,  -  - 
and  0.  ^ 

2.  If  6  =  0,  (1)  becomes  the  pure  quadratic  equation  ax'^  +  c  =  0,  whose 
roots  are  numerically  equal  with  opposite  signs. 

3.  If  c  =  0  and  6  =--  0,  (1)  becomes  ax!^  =  0,  which  has  two  zero  roots. 

4.  If  a  =  0  or  if  a  =  0  and  6  =  0,  (1)  ceases  to  be  a  quadratic  equation. 
But  if  these  coefficients  differ  from  zero,  however  little,  (1)  is  still  a  quad- 
ratic equation  and  has  two  roots.  To  discover  the  nature  of  the  roots  in 
these  cases,  rationalize  the  numerators  in  (2). 

Then,         ri= ^^  and  r^  = — (3) 

-h  -  y/b^  -4ac  ~b  +  Vb^  -  4  ac 

Suppose  that  a  is  very  small  as  compared  with  b  and  c. 
Then,  the  denominator  of  ri  is  very  nearly  equal  to  —  6  —  VP,  or  to  —  2  6, 
and  the  denominator  of  rs  is  very  small. 

Hence,  the  smaller  a  is  the  less  will  the  first  root  differ  from  —  - ,  the 

b 
root  of  the  simple  equation  6x  +  c  =  0,  and  the  greater  will  be  the  numerical 
value  of  the  second  root. 

Suppose  that  a  and  b  are  very  small  as  compared  with  c. 

Then,  both  denominators  in  (3)  are  very  small  as  compared  with  the 
numerators. 

Hence,  the  smaller  a  and  b  are  the  greater  loill  both  roots  be  in  numerical 
value. 

Examples 

1.  What  is  the  nature  of  the  roots  of  a^  —  7a;-8  =  0? 

Solution.  —Since  b"^  -  4:  ac  =  49  +  S2  =  SI  =  92,  the  roots  are  real  and 
unequal  (Prin.  1),  and  rational  (Prin.  2).  Since  c  is  negative,  the  roots  have 
opposite  signs  and,  b  being  negative,  the  positive  root  is  the  greater  numeri- 
cally (Prin.  3). 

2.  What  is  the  nature  of  the  roots  of  Sa^ -{- 5x -^3  =  0? 

Solution.  —  Since  b'^  —  i  ac  =  25  -  36  =  —  11,  both  roots  are  imaginary 
(Prin.  1). 


QUADRATIC  EQUATIONS  315 

Find  the  nature  of  the  roots  of  the  following  equations : 

3.  072  — 5 ic  — 75  =  0.  9.  ar^  +  ic  — 2  =  0. 

4.  a;2-f5a;  +  6  =  0.  10.  4.3?  -  4.x +  !  =  {). 

5.  .T2-j-7a;_30  =  0.  11.  4  a^ +  6  a; -4  =  0. 

6.  a;2-3a;  +  5  =  0.  12.  2a^-9a;  +  4  =  0. 

7.  3^  + 3a;- 5  =  0.  13.  4a.-2  +  16a;  +  7  =  0. 

8.  x2  +  a;  +  2  =  0.  14.  9a;2_^;12a;  +  4  =  0. 

305.   Formation  of  quadratic  equations. 

Any  quadratic  equation,  as  ax^  +  Z>.t  +  c  =  0,  may  be  reduced, 
by  dividing  both  members  by  the  coefB.cient  of  x^,  to  the  form 
a^  -f  jsa;  +  5  =  0,  whose  roots  are 


»  2  2 

Adding  the  roots,     i\  +  ra  =  — — ^  —  —P- 

Z 

Multiplying  the  roots,  r^r^,  —  ^  ~  ^^     — 2i  =  g.     Hence, 

4 

306.  Principle.  —  The  sum  of  the  roots  of  a  quadratic  equa- 
tion having  the  form  oi?  +px  -\-  q  =  0  is  equal  to  the  coefficient  of  x 
with  its  sign  changed,  and  their  product  is  equal  to  the  absolute  term. 

Substituting  —(7\  +  7^)  for  j^,  and  r^r.^  for  q  in  x^  -\- px  -{-  q  =  0, 

«"  —  (^'i  +  r.;)x  +  ri'/-2  =  0. 

Expanding,  a?  —  r^x  —  /-ga;  +  rir^  =  0. 

Factoring,  (x  —  r^)  (x  —  r^  =  0. 

Hence,  to  form  a  quadratic  equation  when  the  roots  are  given  : 

Rule.  —  Subtract  each  root  from  x  and  place  the  product  of  the 
remainders  equal  to  zero. 

Examples 

1.    Form  an  equation  whose  roots  are  —5  and  2. 

Solution,    (a;  +  5)  (x  -  2)  =  0,  or  x^  +  3  a;  -  10  =  0. 

Or,  since  the  sum  of  the  roots  with  their  signs  changed  is  +5  —  2,  or  3,  and 
the  product  of  the  roots  is  —10  (Priu.),  the  equation  is  x^  +  3  x  —  10  =  0. 


316  ACADEMIC  ALGEBRA 

Form  equations  whose  roots  are 


2. 

6,4. 

8. 

a,  —Sa. 

14. 

3+V2,  3-V2. 

3. 

5,  -3. 

9. 

a  4- 2,  a -2. 

15. 

_2-V5,  -2+V5. 

4. 

3,  -i- 

10. 

6  +  1,  ?>_1. 

16. 

2  ±  3Vi. 

5. 

hi- 

11. 

a  +  ?>,  a—  h. 

17. 

-i(3±V6). 

6. 

-2,-i. 

12. 

Va-V6,  V&. 

18. 

i(-l±V2). 

7. 

-i,-f. 

13. 

|(a±V^). 

19. 

a(2±2V5). 

GENERAL  REVIEW 

307.    1 .  Add     X  Vy  +  y  Va;  +  ^'xy,    x^y^  —  Va^V  —  \/xy^,   V^ 
—  Vii?2/^  —  n/^,  and  i/  Vic  —  x  V4^  —  V9  a;?/. 

2.  From  the  sum  of  2a-{-Sh  —  Sy  and  2y  —  a—Sb  sub- 
tract (a  —  6  —  2/)  —  (tt  +  6  H-  ^). 

3.  What  number  must  be  added  to  a  to  give  b  —  a? 

4.  If  a  =  2  and  5  =  3,  find  the  value  of 

a-^2b  b         a^-a^b 

a  a—  b      2a— b 

5.  What  number  must  be  subtracted  from  a  —  6  to  give 
6-a  +  c? 

6.  Simplify  a-;5-a-[a-6-(2a  +  6)  +  (2a-6)-a]-6j. 

7.  A  grocer  sold  m  pounds  of  sugar  at  a  cents  a  pound,  and 
a  pounds  of  tea  at  b  cents  a  pound.  If  the  sugar  cost  him  b  cents 
a  pound  and  the  tea  m  cents  a  pound,  what  was  his  gain  by  the 
transaction  ? 

8.  Multiply  «» +  2  a^d  +  2  a^  J^b^  hj  a^-2  a^b  +  2  aft^  _  b\ 

9.  Multiply  ic"*-!  —  2  2/"-^  by  2a;  +  /. 

10.  Multiply  x-\/x-\-x^y^y^fx-\-y\/y  by  y/x  —  y/y. 

a  a+6  a  0+6 

11.  Multiply  2 a;'-* -5 2/ 2    hy  2  3?^ +  5y~^. 

12.  Expand  (af —  ?/**)  (a;" +  ?/**)  (a^"  +  iy2»^. 

13.  Divide  x^  —  y^  by  a;  — 2/. 


GENERAL  REVIEW  31T 

14.  Divide  ic*  —  3  ar^  —  20  by  x  —  2,  using  detached  coefficients. 

15.  Prove  that  xP^—b^  is  divisible  by  x-\-b. 

16.  Divide  (a  +  b) -{- x  hj  (a -\-  b)^  +  xK 

17.  Factor  9  a^- 12  a; +  4. 

18.  Factor  9x^-{-9x-[-2. 

19.  Factor  aj3-3;^  +  2. 

20.  Prove  that  x  — a  is  sl  factor  of  a;**  +  3  ax"*-^  —  4  a\ 

21.  Separate  d^  —  1  into  six  rational  factors. 

22.  Factor4(ad  +  6c)2-(a2-62_c2-|-(F)2 

23.  Find  the  H.  C.  D.  of  a^-/,  a;^^  2  a^  +  2/^  ajid  y^  +  xy. 

24.  Find  the  H.  C.  D.  of  3a^-a;-2  and  6a^  +  a;-2. 

25.  Find  the  H.  CD.  of  4  a;*- 11  a^  + 11  a;- 12, 

2a^  +  ic3_4a^  +  7a;-15,  and  2  a;*  +  a:"  -  a?  - 12. 

26.  Find  the  L.  C.  M.  of  4  a^bx,  6  abY,  and  2  axy. 

27.  Find  the  L.  C.  M.  of  ar^  —  2/^,  a;  +  y,  and  ajy  —  y\ 

^ 5  a;  4-  6 

28.  Reduce  — ~ —  to  its  lowest  terms. 

a^—  5  a^4-4 

29.  Reduce  — — '      ,~,  to  its  lowest  terms. 

a^-2a^H-l 

rA 1,2 g2 2  fee 

30.  Reduce  — to  its  lowest  terms. 

a2_62  +  c2  +  2ac 

31.  Simplify -^--^^  + 


a;  +  l      1  —  a;     a^—  1 

32.  Simplify -^±^  +  -^^^  +  4^. 

33.  Simplify  ^^  _  j^  (J  _  ^^  -  (^  _  j,^(^  _  „)  +  (^  „) („  _  6)' 

34.  Simplify  («  +  i)(a^  +  i)(a-l). 


35.    Simplify — j -r 

X -—      X'\- 


aJ  +  i  aj-i 

X  X 


318  ACADEMIC  ALGEBRA 


36.    Simplify 


-fl 


1_^1^        x  +  \ 


X 


1-1 


—  X  — 


37.  Prove  that  ^x^  =  ^. 

h      q      hq 

38.  Prove  that  ^-^  =  ^. 

b      71      bm 

39.  Divide:^-^by  J— -L. 

Vy      Vx        Vy     Vx 

40.  Raise  a  —  &  to  the  seventh  power. 

41.  Expand  (2  a -\- 3  by. 

42.  Expand  (V^  +  -v/^)^ 

43.  Square  Va  +  6  —  c. 

44.  Extract  the  square  root  of  a^+2  aVab-\-3  ab+2  bVab-\-b^ 

45.  Extract  the  cube  root  of  8  a^  —  36  a^b  +  54  a^^  _  27  ft^, 

46.  Extract  the  square  root  of  a  +  6  to  four  terms. 

47.  Find  the  sixth  root  of  4826809. 

48.  Reduce  V|  to  its  simplest  form. 

49.  Reduce  \/25  a^  to  its  simplest  form. 

50.  Find  the  approximate  value  of  — . 

-       V2 

51.  Multiply  2  +  V8  by  1  -  V2. 

52.  Simplify  ^  +  ^  A 

V6  +  2 

53.  Prove  that  xO=l. 

54.  Prove  that  ax-'  =  -. 

55.  Prove  that  x^  =  ^',  also  that  x^  =  (^)» 

56.  Find  the  value  of  125^;  of  f^Y^ 

\82j    ' 


GENERAL  REVIEW 


319 


Solve  the  following  equations  : 


69. 
60. 
61, 


68. 


69. 


70. 


fl. 


57. 


58. 


a  —  h 

1 

X 

a 


+ 


a  —  h 

X 


X  — 


a  +  6 

1 
X 


+ 


a  +  h 


+ 


+ 


=  0. 


ma?  —  nx  =  mn. 

0^  +  1-5^. 

"^^2-   2 


63.    Va;-9=V^-1. 


x^^S  =  9a?. 
62.    (l-{-xy-\-(l-xy=2^2. 
14- a; 


64.    ar^+vV  +  16  =  14. 
65. 


^-\-x]  -(--\-x]  =  20. 


66. 


l  +  ic+vTT^ 


72 


73. 


74. 


67. 

x-^y  =  H, 
y-\-z=:4:, 
z  +  x  =  6. 

ic     y 

?  +  ?  =  10. 

aj     2/ 

0?  + 2^4- 32  =  13, 
.Sx-\-  y  +  2z  =  ll. 

'  ax-^y-\-z  =  2{a  +  l), 
x-\- ay -\-z  =  Sa-\- ly 
.  X  -\-  y  -\-  az  =  a^  +  S. 

x^  -\-  xy  =  24t, 
-{-xy  =  12. 

x2  +  3  a^  =  7, 
0^  +  42/2  =  18. 

x^y  +  xy'^  =  6, 
a^  -f-  2/3  ^  9^ 


=  a 


a;        J       \x 

a^  +  a^+(l  +  a;  +  aj2)2=55. 
1  +  a; 


75. 


76. 


77. 


78. 


79. 


80. 


81. 


82. 


83. 


84. 


l-a;+Vl+a5* 
ar^  +  a;  =  26  -  /  -  2/, 
xy  =  S. 

Vxy  =  12, 


x  +  y—-Vx-\-y=:20. 

2/^  =  7, 

2/*  =  175. 
xy-xy^  =  -6, 
I  a;  —  xy^  =  9. 

a^  =  a;  +  2/> 
a^  +  2/'  =  8. 
(a?y^-4:xy  =  5, 

I  3^  +  42/^^  =  29. 
f2x-3  +  22/3_9^y^ 

a?  +  2/  =  3. 

^  +  a^  2/'  -  189, 
.  X  +  Va:2/  +  y  =  21. 
I  a)^  +  2/^  =  4, 
U'^  +  2/  =  16. 

I  V^-  V2^  =  f  (a;  -  2/), 
1  V«2/  =  -J-. 


320  ACADEMIC  ALGEBRA 

85.  A  and  B  hired  a  carriage  for  themselves  and  four  friends. 
If  all  had  paid,  A  and  B  would  each  have  had  4  dollars  less  to 
pay.     What  was  the  cost  of  hiring  the  carriage  ? 

86.  What  number  is  that  to  which  if  12  is  added  and  from 
^  of  the  sum  12  is  subtracted,  the  remainder  is  12  ? 

87.  A  grocer  has  two  kinds  of  sirup  worth  50  and  80  cents 
per  gallon  respectively.  How  many  gallons  of  each  must  he  take 
to  make  a  mixture  of  45  gallons  worth  60  cents  a  gallon  ? 

88.  How  many  dimes  and  how  many  quarters  must  be  taken 
so  that  18  coins  are  worth  $S? 

89.  In  a  certain  weight  of  gunpowder  the  saltpeter  was  5 
pounds  more  than  half  the  weight,  the  sulphur  2  pounds  less 
than  a  fifth,  and  the  charcoal  1  pound  more  than  a  tenth.  Find 
the  number  of  pounds  of  each. 

90.  How  far  down  a  river  whose  current  runs  3  miles  an  hour 
can  a  steamboat  go  and  return  in  8  hours,  if  its  rate  of  sailing  in 
still  water  is  12  miles  an  hour  ? 

91.  A  woman  being  asked  what  she  paid  for  her  eggs,  replied, 
"Six  dozen  cost  as  many  cents  as  I  can  buy  eggs  for  32  cents." 
What  was  the  price  per  dozen  ? 

92.  A  gentleman  had  not  room  in  his  stables  for  8  of  his 
horses,  so  he  built  an  additional  stable  ^  the  size  of  the  other, 
and  then  had  room  for  8  horses  more  than  he  had.  How  many 
horses  had  he  ? 

93.  In  a  mass  of  copper,  lead,  and  tin,  the  copper  was  5 
pounds  less  than  half  the  whole  in  weight,  and  the  lead  and  tin 
each  5  pounds  more  than  ^  of  the  remainder.  Find  the  weight 
of  each. 

94.  At  what  time  between  4  and  5  o'clock  do  the  hands  of 
a  clock  make  a  straight  line  ? 

95.  A  person  who  can  walk  n  miles  an  hour  has  a  hours  at  his 
disposal.  How  far  may  he  ride  in  a  coach  that  travels  m  miles 
an  hour  and  return  on  foot  within  the  allotted  time  ? 


GENERAL   REVIEW  321 

96.  A  merchant  sold  half  a  car  load  more  than  half  his  grain ; 
then  he  sold  half  a  car  load  more  than  half  the  remainder,  when 
he  found  that  if  he  could  sell  half  a  car  load  more  than  half  of 
what  he  still  had,  he  would  have  none  left.  How  many  car 
loads  of  grain  had  he  ? 

^  97.    Four  years  ago  A's  age  was  \  of  B^s,  and  4  years  hence 
it  will  be  I  of  B's  age.     What  is  the  age  of  each  ? 

98.  A  person  being  asked  the  time  of  day  replied  that  the 
time  past  noon- was  f  of  the  time  to  midnight.  What  was  the 
time  of  day  ? 

99.  If  3  is  added  to  each  term  of  a  certain  fraction,  the  value 
of  the  fraction  will  be  f ;  if  3  is  subtracted  from  each  term,  the 
value  will  be  4.     What  is  the  fraction  ? 

100.  A  boatman  rows  such  a  distance  down  a  stream  that  it 
takes  him  4  hours  to  return.  If  it  takes  him  2  hours  to  row 
down  and  the  current  is  2  miles  an  hour,  what  is  his  rate  of 
rowing  in  still  watei;? 

101.  A  man  received  $  2.50  per  day  for  every  day  he  worked, 
and  he  agreed  to  forfeit  $  1.50  for  every  day  he  was  idle.  If  he 
worked  3  times  as  many  days  as  he  was  idle  and  received  $  24, 
how  many  days  did  he  work  ? 

102.  A  jeweler  has  two  silver  cups,  and  a  cover  worth  $  1.50. 
The  first  cup  with  the  cover  on  it  is  worth  1\  times  as  much  as 
the  second  cup,  and  the  second  cup  with  the  cover  on  it  is  worth 
\^  as  much  as  the  first  cup.     Find  the  value  of  each  cup. 

103.  Some  smugglers  discovered  a  cave  that  would  exactly 
hold  their  cargo,  which  consisted  of  13  bales  of  cotton  and  33 
casks  of  wine.  While  they  were  unloading,  a  revenue  cutter 
hove  in  sight,  and  they  sailed  away  with  9  casks  and  5  bales, 
leaving  the  cave  two  thirds  full.  How  many  bales,  or  how  many 
casks,  would  the  cave  hold  ? 

104.  Twenty-eight  tons  of  goods  are  to  be  transported  in  carts 
and  wagons,  and  it  is  found  that  it  will  require  15  carts  and  12 
wagons,  or  else  24  carts  and  8  wagons.  How  much  can  each  cart 
and  each  wagon  carry  ? 

ACAD.    ALG.  21 


322  ACADEMIC  ALGEBRA 

105.  There  is  a  number  whose  three  digits  are  the  same.  If  7 
times  the  sum  of  the  digits  is  subtracted  from  the  number,  the 
remainder  is  180.     What  is  the  number  ? 

106.  A  and  B  can  do  a  piece  of  work  in  m  days,  B  and  C  in  n 
days,  A  and  C  in  p  days.  In  what  time  can  all  together  do  it? 
How  long  will  it  take  each  alone  to  do  it  ? 

107.  Two  passengers  together  have  400  pounds  of  baggage 
and  are  charged,  for  the  excess  above  the  weight  allowed  free,  40 
and  60  cents  respectively.  If  the  baggage  had  belong:ed  to  one 
of  them,  he  would  have  been  charged  $  1.50.  How  much  baggage 
is  one  passenger  allowed  without  charge  ? 

108.  Divide  20  into  two  parts  such  that  the  sum  of  the  two 
fractions  formed  by  dividing  each  part  by  the  other  is  4 J. 

109.  It  takes  1000  square  tiles  of  a  certain  size  to  pave  a  hall, 
or  1440  square  tiles  whose  dimensions  are  one  inch  less.  Find 
the  area  of  the  hall  floor. 

110.  The  sum  of  two  numbers  is  16,  and  the  difference  of  their 
squares  is  128.     What  are  the  numbers  ? 

111.  Find  two  numbers  such  that  their  sum,  their  product,  and 
the  difference  of  their  squares  are  all  equal. 

112.  Divide  25  into  two  parts  such  that  the  difference  of  their 
square  roots  is  1. 

113.  The  difference  of  two  numbers  is  6,  and  their  product 
is  equal  to  twice  the  cube  of  the  less  number.  What  are  the 
numbers  ? 

114.  It  took  a  number  of  men  as  many  days  to  pave  a  side- 
walk as  there  were  men.  Had  there  been  3  men  more,  the  work 
would  have  been  done  in  4  days.     How  many  men  were  there  ? 

115.  The  product  of  two  numbers  is  8,  and  the  sum  of  their 
squares  is  14  greater  than  the  sum  of  the  numbers.  What  are 
the  numbers  ? 

116.  A  rectangular  lawn  50  feet  long  and  40  feet  wide  has  a 
walk  of  uniform  width  around  it.  If  the  area  of  the  walk  is  64 
square  yards,  what  is  its  width  ? 


GENERAL  REVIEW  323 

117.  A  merchant  sold  goods  for  56  dollars  and  gained  as  many 
hundredths  of  the  cost  as  there  were  dollars  in  the  cost.  Find 
the  cost  of  the  goods. 

118.  A  person  swimming  in  a  stream  that  runs  1^  miles  per 
hour  finds  that  it  takes  him  3  times  as  long  to  swim  a  certain 
distance  up  the  stream  as  it  does  to  swim  the  same  distance  down. 
What  is  his  rate  of  swimming  in  still  water  ? 

119.  A  drover  bought  some  oxen  for  $  900.  After  5  had  died, 
he  sold  the  rest  at  a  profit  of  $  20  each  and  thereby  gained  $  350. 
How  many  oxen  did  he  buy  ? 

120.  A  detachment  from  an  army  was  marching  in  regular 
column  with  5  men  more  in  depth  than  in  front.  On  approaching 
the  enemy,  the  front  was  increased  by  845  men,  and  the  whole 
was  thus  drawn  up  in  5  lines.  Find  the  number  of  men  in  the 
detachment. 

121.  A  round  iron  bar  weighed  36  pounds.  If  it  had  been 
1  foot  longer  and  of  uniform  diameter,  each  foot  of  it  would  have 
weighed  \  a  pound  less.  Find  the  length  of  the  iron  bar  and  its 
weight  per  foot. 

122.  A  farmer  has  two  cubical  granaries.  The  side  of  one  is 
3  yards  longer  than  the  side  of  the  other,  and  the  difference  of 
their  solid  contents  is  117  cubic  yards.  What  is  the  length  of 
the  side  of  each  ? 

123.  Two  workmen,  A  and  B,  were  employed  at  different 
wages.  At  the  end  of  a  certain  number  of  days  A  received  $30, 
but  B,  who  had  been  idle  two  days  in  the  meantime,  received 
only  $  19.20.  If  B  had  worked  the  whole  time,  and  A  had  been 
idle  two  days,  they  would  have  received  equal  sums.  Find  the 
number  of  days,  and  the  daily  wages  of  each. 

124.  By  traveling  5  miles  an  hour  less  than  its  usual  rate  a 
train  was  50  minutes  late  in  running  300  miles.  Find  the  usual 
rate  of  speed  and  the  time  usually  required  to  make  the  trip. 

125.  Find  two  numbers  such  that  their  sum,  their  product,  and 
the  sum  of  their  squares  are  all  equal. 


324  ACADEMIC  ALGEBRA 

126.  A  inerchant  bought  two  lots  of  tea,  paying  for  both  $34. 
One  lot  was  20  pounds  more  than  the  other,  and  the  number  of 
cents  paid  per  pound  was  in  each  case  equal  to  the  number  of 
pounds  bought.     How  many  pounds  of  each  did  he  buy  ? 

127.  A  and  B  hired  a  pasture  into  which  A  put  4  horses,  and 
B  as  many  as  cost  him  18  shillings  per  week.  Afterward  B  put 
in  2  additional  horses,  and  found  that  he  must  pay  20  shillings 
per  week.     How  much  was  paid  for  the  pasture  per  week  ? 

128.  By  lowering  the  selling  price  of  apples  1  cent  a  dozen, 
an  apple  woman  finds  that  she  can  sell  60  more  than  she  used  to 
sell  for  60  cents.  At  what  price  per  dozen  did  she  sell  them  at 
first? 

129.  A  and  B  are  two  stations  300  miles  apart.  Two  trains 
start  at  the  same  time,  one  from  A,  the  other  from  B,  and  travel 
to  the  opposite  station.  If  the  first  train  reaches  B  9  hours  after 
the  trains  meet,  and  the  second  train  reaches  A  4  hours  after 
they  meet,  when  do  they  meet,  and  what  is  the  rate  of  each  train  ? 

• 

130.  If  a  carriage  wheel  14|  feet  in  circumference  takes  one 

second  longer  to  revolve,  the  rate  of  traveling  will  be  2 J  miles 
less  per  hour.     How  fast  is  the  carriage  traveling  ? 

131.  A  railway  train,  after  traveling  2  hours,  was  detained  1 
hour  by  an  accident.  It  then  proceeded  at  f  of  its  former  rate, 
and  arrived  7f  hours  behind  time.  If  the  accident  had  occurred 
50  miles  farther  on,  the  train  would  have  arrived  6J  hours  behind 
time.     What  was  the  whole  distance  traveled  by  the  train  ? 

132.  A  person  rents  a  certain  number  of  acres  of  land  for 
$  200.  He  retains  5  acres  for  his  own  use  and  sublets  the  rest 
at  $  1  an  acre  more  than  he  gave.  If  he  receives  $  10  more  than 
he  pays  for  the  whole,  how  many  acres  does  he  rent,  and  at  what 
rate  per  acre  ? 

133.  A  and  B  left  Chicago  and  walked  in  the  same  direction 
at  uniform  rates.  B  started  2  hours  after  A  and  overtook  him 
at  the  30th  milestone.  Had  each  traveled  half  a  mile  more  per 
hour,  B  would  have  overtaken  A  at  the  42d  milestone.  At  wl\at 
rate  did  each  travel  ? 


RATIO    AND   PROPORTION 


308.  1.  What  is  the  relation  of  10  a;  to  5  x ?  oiSxtol2x?  of 
Sato  2  a?  of  Um  to  7  m?  of  4a  to  8a?  of  26  to  66? 

2.  In  finding  the  relation,  or  ratiOy  of  10  a  to  5  a,  which  is  the 
dividend,  the  number  that  j^rececZes,  or  the  number  that  follows  ? 
Which  is  the  divisor  ? 

3.  What  is  the  ratio  of  a  to  6  ?  Since  6  may  not  be  exactly- 
contained  in  a,  how  may  the  ratio  be  expressed  ? 

4.  Since  the  ratio  of  two  numbers  may  be  expressed  in  the 
form  of  a  fraction,  what  operations  may  be  performed  upon  the 
terms  of  a  ratio  without  changing  the  ratio  ? 

309.  The  relation  of  two  numbers  that  is  expressed  by  the 
quotient  of  the  first  divided  by  the  second  is  called  their  Ratio. 

310.  The  Sign  of  Ratio  is  a  colon  (:). 

A  ratio  is  also  expressed  in  the  form  of  a  fraction. 

The  ratio  of  a  to  b  is  written  a  :  6  or  -• 

b 
The  colon  is  sometimes  regarded  as  derived  from  the  sign  of  division  by 
omitting  the  line. 

311.  The  first  term  of  a  ratio  is  called  the  Antecedent. 
It  corresponds  to  a  dividend,  or  numerator. 

312.  The  second  term  of  a  ratio  is  called  the  Consequent. 
It  corresponds  to  a  divisor,  or  denominator. 


313.    The  antecedent  and  consequent  form  a  Couplet 

In  the  ratio  a  :  b,  or  -,  a 
b 

terras  a  and  6  form  a  couplet. 


In  the  ratio  a  :  &,  or  -,  a  is  the  antecedent,  b  the  consequent,  and  the 
b 


325 


326  ACADEMIC  ALGEBRA 

314.  The  ratio  of  the  reciprocals  of  two  numbers  is  called  the 
Reciprocal,  or  Inverse  Ratio   of  the  numbers. 

It  may  be  expressed  by  interchanging  the  terms  of  the  ratio 
of  the  numbers. 

The  inverse  ratio  of  a  to  6  is  -  :     .     Since    --=--  =  -,  the  inverse  ratio  of 


6 


a    b  aba 


a  to  6  may  he  written     ,  or  b  la. 
a 

315.  The  ratio  of  the  squares  of  two  numbers  is  called  the 
Duplicate  ratio ;  the  ratio  of  their  cubes,  the  Triplicate  ratio ;  the 
ratio  of  their  square  roots,  the  Subduplicate  ratio;  the  ratio  of 
their  cube  roots,  the  Subtriplicate  ratio  of  the  numbers. 

The  duplicate  ratio  of  a  to  6  is  a^  .  52 .  the  tripHcate  ratio,  a^  :  b^ ;  the 
subduplicate  ratio,  Va:Vb;  the  subtriplicate  ratio,  \/a  :  Vb. 

316.  Principle.  — Multiplying  or  dividing  both  terms  of  a  ratio 
by  the  same  number  does  not  change  the  ratio. 

Examples 

1.   What  is  the  ratio  of  8  m  to  4  m  ?  of  4  m  to  8  m  ? 

2c  Express  the  ratio  of  6:9  in  its  lowest  terms;  12 a?:  16?/; 
am  \bm\  20  a6  :  10  6c ;  (m  +  n)  :  (m^  —  t?), 

3.  Which  is  the  greater  ratio,  2:3  or  3:4?  4:9  or  2:5? 

4.  What  is  the  ratio  of  |  to  J  ?  i  to  |  ?  |  to  f  ? 

Suggestion.  —  When  fractions  have  a  common  denominator,  they  have 
the  ratio  of  their  numerators. 

5.  Reduce  a :  b  and  xiy  to  ratios  having  the  same  consequent. 

6.  When  the  antecedent  is  6  x  and  the  ratio  is  i,  what  is  the 
consequent  ? 

317.  It  is  evident  from  §  316,  that  the  ratio  of  two  rational 
fractions  may  be  expressed  by  the  ratio  of  two  integers. 

For  example,  —  :  _  may  be  reduced  to  the  form  —  xnyi-xny, 
7  n     y  n  V 

or  my :  bn.  ^  '^  y 

But  the  ratio  of  two  numbers,  when  one  is  rational  and  the 
other  irrational  or  when  they  are  dissimilar  surds,  cannot  be 
expressed  by  the  ratio  of  two  integers. 

Thus,  the  ratio  V2  :  3  cannot  be  expressed  by  any  two  integers. 


RATIO   AND  PROPORTION  327 

318.  If  the  ratio  of  two  numbers  can  be  expressed  by  the  ratio 
of  two  integers,  the  numbers  are  called  Commensurable  Numbers, 
and  their  ratio  a  Commensurable  Ratio. 

319.  If  the  ratio  of  two  numbers  cannot  be  expressed  by  the 
♦ratio  of  two  integers,  the  numbers  are  called  Incommensurable 

Numbers,  and  their  ratio  an  Incommensurable  Ratio. 

The   ratio    V2  :  3  =  — -  =  — — -— ^ —  cannot  be  expressed   by  any  two 
o  o 

integers,  because  there  is  no  number  that,  used  as  a  common  measure^  will 
be  contained  in  both  y/2  and  3  an  integral  number  of  times.  Hence,  V2  and 
3  are   incommensurable,  and  \/2  :  3  is  an  incommensurable  ratio. 

It  is  evident  that  by  continuing  the  process  of  extracting  the  square  root 
of  2,  the  ratio  ■\/2  : 3  may  be  expressed  by  two  integers  to  any  desired  de- 
gree of  approximation,  but  never  with  absolute  accuracy. 

320.  1.  What  two  numbers  have  the  same  relation  to  each 
other  as  2  to  3  ? 

2.  Name  several  couplets  that  express  the  same  ratio  as  2:5. 
How  may  it  be  indicated  that  the  ratio  of  2  to  5  is  the  same  as 
that  of  2  a  to  5  a  ? 

3.  What  number  has  the  same  ratio  to  12  a  that  5  h  has  to  3  6  ? 

4.  What  number  has  the  same  ratio  to  10  a  that  10  a  has  to  2  ? 
How  does  this  number  times  2  compare  with  10  a? 

321.  An  equality  of  ratios  is  called  a  Proportion. 

3  :  10  =  6  :  20  and  a  :  x  =  6  :  y  are  proportions. 

The  double  colon  (:  :)  is  often  used  instead  of  the  sign  of 
equality. 

The  double  colon  has  been  supposed  to  represent  the  extremities  of  the 
lines  that  form  the  sign  of  equality. 

The  proportion  a  :  6  =  c  :  d,  or  a  :  6  :  :  c  :  c?,  is  read,  the  ratio  of  a 
to  h  is  equal  to  the  ratio  of  c  to  d,  or  a  is  to  6  as  c  is  to  d. 

322.  The  antecedents  and  consequents  of  the  ratios  that  form 
a  proportion  are  called  the  Antecedents  and  Consequents,  respec- 
tively, of  the  proportion. 

In  a  :  h  =  c. :  d^  the  antecedents  of  the  proportion  are  a  and  c,  and  the 
consequents  are  b  and  d. 


328  ACADEMIC  ALGEBRA 

323.  The  first  and  fourth  terms  of  a  proportion  are  called  the 
Extremes  of  the  proportion. 

In  the  proportion  a  :  b  =  c  :  d,  the  extremes  are  a  and  d. 

324.  The  second  and  third  terms  of  a  proportion  are  called  the 
Means  of  the  proportion. 

In  the  proportion  a  :b  =  c  :d,  the  means  are  b  and  c. 

325.  The  terms  of  a  proportion  are  also  called  Proportionals. 
In  the  proportion  a:  b  =  b  :c,  b  is  called  a  Mean  Proportional 
between  a  and  c,  and  c  is  called  a  Thiy^d  Proportional  to  a  and  h. 

In  the  proportion  a :  6  =  c :  d,  d  is  called  a  Fourth  Proportional 
to  a,  6,  and  c. 

Since  a  proportion  is  an  equality  of  ratios  each  of  which  may 
be  expressed  as  a  fraction,  a  proportion  may  be  expressed  as  an 
equation  each  member  of  which  is  a  fraction.  Hence,  it  follows 
that: 

326.  General  Principle.  —  The  changes  that  may  he  made 
upon  a  proportion  without  destroying  the  equality  of  its  ratios  are 
based  upon  the  chayiges  that  may  be  made  upon  the  members  of  an 
equation  without  destroying  their  equality  and  upon  the  terms  of 
a  fraction  without  altering  the  value  of  the  fraction. 

PRINCIPLES   OP   PROPORTION 

327.  1.  Let  any  four  numbers  form  a  proportion,  as  a  :  6  =  c :  do 

2.  Express  the  proportion  as  a  fractional  equation. 

3.  If  this  equation  is  cleared  of  fractions,  what  terms  of  the 
proportion  does  the  first  member  contain  ?  the  second  member  ? 

Principle  1.  —  In  any  proportion  the  product  of  the  extremes 
is  equal  to  the  product  of  the  means. 

If  a:b  =  c  :d,  then,  ad  =  be. 

Since  a  mean  proportional  serves  as  both  means  of  a  proportion, 
if  a:b  =  b:Cj  b^  =  ac,  ov  b  =  ■\/ac.     Hence, 

Tlie  mean  proportional  between  two  numbers  is  equal  to  the  square 
root  of  their  product. 


RATIO  AND  PROPORTION  329 

Principle  1  may  be  established  as  follows : 
Let  a  :  6  =  c  :  d  represent  any  proportion. 

Then,  §  310,  2  =  -. 

h      d 

Clearing  of  fractions,  ad  =  be. 

Therefore,  the  product  of  the  extremes  is  equal  to  the  product  of  the 
means. 

Numerical  Illustration 

2  :  5  =  6  :  15. 

2  X  15  =  5  X  6. 

30  =  30. 

328.  1.  Transform  the  proportion  a:b  =  c:d  in  accordance 
with  Prin.  1. 

2.  Since  ad  =  be,  how  may  the  value  of  a  be  found  ?  the  value 
ot  d?     What  terms  of  the  proportion  are  a  and  d  ? 

3.  How,  then,  may  either  extreme  of  a  proportion  be  found  ? 
How  may  either  mean  be  found  ? 

Principle  2.  —  Either  extreme  of  a  proportion  is  equal  to  the 
product  of  the  means  divided  by  the  other  extreme.  Either  mean  is 
equal  to  the  product  of  the  extremes  divided  by  the  other  mean. 

K  a  :b  =  c:di  then,  a  =  — ,   6  =  — ,  etc. 

d  c 

Demonstrate  Prin.  2,  and  give  numerical  illustrations. 

329.  1.  If  ad=zbCy  what  will  be  the  resulting  proportion  when 
both  members  are  divided  by  bd  and  reduced  ? 

2.  What  do  the  factors  of  ad,  the  first  member  of  the  equation, 
form  in  the  proportion  ?     What  do  the  factors  of  be  form  ? 

Principle  3.  —  If  the  product  of  two  numbers  is  equal  to  the 
product  of  two  other  numbers,  one  pair  of  them  may  be  made  the 
extremes  and  the  other  pair  the  mearis  of  a  proportioyi. 

If  ad  =  be,  then,  a  .b  —  c  id,  or  b  •.a  =  d:  c,  etc. 

Demonstrate  Prin.  3,  and  give  numerical  illustrations. 


330  ACADEMIC  ALGEBRA 

330.  1.  Change  the  proportion  a:h  =  cf-.d  into  an  integral 
equation  by  Frin.  1. 

2.  Divide  the  members  of  this  equation  by  cd  and  reduce. 

3.  What  terms  of  the  given  proportion  now  form  the  first 
couplet  ?   the  second  couplet  ? 

Principle  4.  —  If  four  numbers  are  in  proportion,  the  ratio 
of  the  antecedents  is  equal  to  the  ratio  of  the  consequents  ;  that  is, 
the  numbers  are  in  proportion  by  Alternation. 

If  a-.h  =  c:d^  then,  a:c=h:d. 

Demonstrate  Prin.  4,  and  give  numerical  illustrations. 

331.  1.  Change  the  proportion  a:b  =  c:d  into  an  integral 
equation  by  Prin.  1. 

2.  Divide  the  members  of  this  equation,  be  =  ad,  by  ac,  and 
reduce. 

3.  What  change  has  taken  place  in  the  order  of  the  terms  of 
each  couplet  ? 

Principle  5.  —  If  four  numbers  are  in  propoHion,  the  ratio 
of  the  second  to  the  first  is  equal  to  the  ratio  of  the  fourth  to  the 
third  ;  that  is,  the  numbers  are  in  proportion  by  Inversion. 

If  a  '.  h  =  c  :  d,  then,  b  :  a  =  d  :  c. 

Demonstrate  Prin.  5,  and  give  numerical  illustrations. 

332.  1.  Express  the  proportion  a:b  =  c:d  as  a  fractional 
equation. 

2.  Add  1  to  each  member  and  reduce  the  mixed  numbers  to 
fractional  form.     Write  in  the  form  of  a  proportion. 

3.  How  may  the  terms  of  this  proportion  be  formed  from  the 
terms  of  the  given  proportion  ? 

4.  Since,  when  a  :  b  =  c:  d,  b  :  a  =  d:  c,  if  the  changes  just 
indicated  are  made  in  the  second  proportion,  how  may  the  terms 
of  the  resulting  proportion  be  obtained  from  the  terms  of  the' 
original  proportion? 


RATIO  AND  PROPORTION  8S1 

Principle  6.  —  If  four  numbers  are  in  proportion,  the  sum 
of  the  terms  of  the  first  ratio  is  to  either  term  of  the  first  ratio  as 
the  sum  of  the  terms  of  the  second  ratio  is  to  the  corresponding 
term  of  the  second  ratio  ;  that  is,  the  numbers  are  in  proportion  by 
Composition. 

If  a  :  6  =  c  :  (?,  then,  a  +  h  ih  =  c  -\-  d  id  and  a  -{-  b  :  a  =  c  -{■  d:  c. 
Demonstrate  Prin.  6,  and  give  numerical  illustrations. 

333.  1.  Express  the  proportion  a  :  6  =  c :  c?  as  a  fractional 
equation. 

2.  Subtract  1  from  each  member,  and  reduce  the  mixed  num- 
bers to  fractional  form.     Write  in  the  form  of  a  proportion. 

3.  How  may  the  terms  of  this  proportion  be  formed  from  the 
terms  of  the  given  proportion  ? 

4.  Since,  when  a:b  =  c:d,  b  :  a  =  d  :  Cj  if  the  changes  just 
indicated  are  made  in  the  second  proportion,  how  may  the  terms 
of  the  resulting  proportion  be  obtained  from  the  terms  of  the 
original  proportion  ? 

Principle  7.  —  If  four  numbers  are  in  proportion,  the  differ- 
ence  between  the  terms  of  the  first  ratio  is  to  either  term  of  the 
first  ratio  as  the  difference  between  the  terms  of  the  second  ratio  is 
to  the  corresponding  term  of  the  second  ratio;  that  is,  the  numbers 
are  in  proportion  by  Division. 

If  a  :  &  =  c  :  (?,  then,  a  —  h  :h  =  c  —  d  id.  and  a  -h  :a  =  c  —  d  ic. 

Demonstrate  Prin.  7,  and  give  numerical  illustrations. 

334.  1.  Change  the  proportion  a:  b  =  c:  d  according  to  Prin. 
6,  and  also  according  to  Prin.  7,  using  the  same  consequents  in 
each  transformation.     Express  in  fractional  form. 

2.  Divide  the  first  equation  by  the  second. 

3.  How  may  the  terms  of  the  resulting  proportion  be  formed 
from  the  terms  of  the  given  proportion  ? 

Principle  8.  —  If  four  numbers  are  in  proportion,  the  sum  of 
the  terms  of  the  first  ratio  is  to  their  difference  as  the  sum  of  the 


332  ACADEMIC  ALGEBRA 

terms  of  the  second  ratio  is  to  their  difference;  that  is,  the  numbers 
are  in  proportion  by  Composition  and  Division. 

If  a  :  6=c:  d^  a+&  :  a  —  h  =  c  +  d  :  c  —  d  and  a  +  b  :  b  —  a  =  c+d  :  d—c. 

Demonstrate  Prin.  8,  and  give  numerical  illustrations. 

335.  1.  Express  the  proportion  a:b  =  c:d  as  a  fractional 
equation. 

2.  Raise  each  member  to  the  nth  power. 

3.  Also  express  the  nth.  root  of  each  member. 

4.  How  may  these  proportions  be  formed  from  the  given 
proportion  ? 

Principle  9.  —  If  four  numbers  are  in  proportion,  their  like 

powers,  and  also  their  like  roots,  will  be  in  proj^ortion. 

2     12     1 
If  a  :  6  =  c  :  d,  then,  a'':b''  =  c»:  d«  and  a"  :  6«  =  c«  :  d». 

Demonstrate  Prin.  9,  and  give  numerical  illustrations. 

336.  1.  Express  a  :  6  =  c :  d  as  a  fractional  equation. 

2.  What  may  be  done  to  a  fraction  without  changing  its  value  ? 

3.  Multiply  the  terms  of  the  first  fraction  by  m  and  the  terms 
of  the  second  fraction  by  n.     Write  as  a  proportion. 

4.  How  may  the  terms  of  this  proportion  be  formed  from  the 
terms  of  the  given  proportion  ? 

5.  Take  the  given  proportion  by  alternation  and  multiply  the 
terms  of  the  first  couplet  by  m  and  those  of  the  second  couplet 
by  n. 

6.  How  may  the  terms  of  this  proportion  be  formed  from  the 
terms  of  the  given  proportion  ? 

7.  How  may  the  given  proportion  be  formed  from  the  propor- 
tions ma  :mb  =  nc:nd  and  ma  :  7ib  =  mc  :  nd  ? 

Principle  10.  —  In  a  proportion,  if  both  terms  of  a  couplet,  or 
both  antecedents,  or  both  consequents  are  multiplied  or  divided  by 
the  same  number,  the  resulting  four  numbers  form  a  proportion. 

If  a'.b  =  C'.d,  then,  ma  :  mh  =  nc -.  nd  and  ma  :  nb  =  mc -.  nd  \  also,  if 
ma  :mb  =  nc  :  nd,  or  if  ma  :  nb  =  mc  i  nd,  then,  a  :b  =  c:  d. 

Demonstrate  Prin.  10,  and  give  numerical  illustrations. 


RATIO  AND  PROPORTION  333 

337.  1.  Express  the  proportions  a:h  =  c:d  and  x:y  —  z:w 
as  fractional  equations. 

2.  How  may  two  equations  be  combined  (Ax.  4  and  5)  ?  Com- 
bine these  two  equations  and  write  the  results  as  proportions. 

3.  How  may  these  proportions  be  formed  from  the  given 
proportions  ? 

Principle  11.  —  The  products,  and  also  the  quotients,  of  corre- 
sponding terms  of  two  proportions  foi'm  a  proportion. 

It  a  .  0  =  c  :d  and  x  :  y  —  z  :  Wj  ax  :by  =  cz  :  dw,  also  ^  :-  =  £:_ . 

X    y       z    w 

Demonstrate  Prin.  11,  and  give  numerical  illustrations. 

338.  If  a:b  =  c:d  and  c:d  =  e:f,  how  does  the  ratio  a :  b 
compare  in  value  with  the  ratio  e:f? 

Principle  12. — If  two  proportions  have  a  common  couplet,  the 
other  two  couplets  will  form  a  proportion. 

If  a:b  =  c:d  and  c:d  =  e:f^  then,  a:b  =  e  :f. 

Demonstrate  Prin.  12,  Und  give  numerical  illustrations  (Ax.  1). 

339.  A  proportion  that  consists  of  three  or  more  equal  ratios 
is  called  a  Multiple  Proportion. 

2:4  =  3:6  =  5:  10  and  a  :b  ^  c:d  =  e  :f  are  multiple  proportions. 

\ 
^  340.    A  multiple  proportion  in  which  each  consequent  serves 

also  as  the  antecedent  of  the  following  ratio  is  called  a  Continued 

Proportion. 

2  : 4  =  4  :  8  =  8  :  16  and  a:b  =  b:c  =  c:d  are  continued  proportions. 

341.   1.  Form  a  multiple  proportion,  as 

2:4  =  3:6  =  5:10  =  10:20. 

2.  How  does  the  ratio  of  the  sum  of  the  antecedents  to  the  sum 
of  the  consequents  compare  with  the  first  ratio  ?  with  the  second 
ratio  ?   with  the  ratio  of  any  antecedent  to  its  consequent  ? 

3.  Investigate  other  multiple  proportions. 


334  ACADEMIC  ALGEBRA 

^Principle  13.  —  In  any  multiple  proj)07'tion  the  sum  of  the  ante- 
cedents is  to  the  su7)i  of  the  consequents  as  any  antecedent  is  to  its 
consequent. 

Principle  13  may  be  established  as  follows : 

Let  a  :b  =  c:d  =  e  :f=  g  :h. 

It  is  to  be  proved  that 

a-\-c  +  e  +  g:b-^d-\-f+h  =  a:by  or  c:d,  etc. 
Denoting  each  of  the  equal  ratios  by  r, 

Hence,  a  =  br,   c  =  dr,   e=fr,  g  =  hr.  (2) 

Adding  equations  (2), 

a  +  c-{-e-{-g=(b  +  d  +  f-\-h)r.  (3) 

Dividing  hy  (b  +  d  +  f-\^  h), 

a-hc+e  +  g  ^^ 
b  +  d+f-Yh 

Replacing  r  by  any  of  the  equal  ratios, 

a^c-^e  +  g  ^a^c   ^^^ 
b-{-d^-f+h      b     d' 

That  is,     a  +  c  ^-  e  -\-  g  -.b  -\-  d  -\-  f  -\-  h  =  (\:b,  ov  c-.d^  etc. 

Examples 
342.    1.  In  the  proportion  3  :  5  =  x  :  55,  what  is  the  value  of  x? 


First  Solution 

3  :  6  =  X  :  55. 

Prin.  2, 

x=^-^^  =  33. 
5 

Second  Solution 

3  :  5  =  X  :  55. 

Prin.  10, 

3:l  =  a;:ll. 

Prin.  1, 

X  =  33. 

Find  the  value  of  x  in  each  of  the  following  proportions : 

2.  2:  3  =  6:  a;.  4.    l:a;  =  a;:9. 

3.  5:a;  =  4:3.  6     8  :  5  =  a; :  10. 


RATIO  AND  PROPORTION  335 

6.  x  +  l:x  =  ^:Q.  '    8.    x  +  2:a;  =  10:6. 

7.  a;:aj-l  =  15:12.  9.    x-^2  :  x-2  =  ^-.1. 
-  10.    If  a;  -j-  5  :  a;  —  5  =  5  :  3,  find  the  value  of  x. 

11.  What   two   numbers   are   mean   proportionals   between   1 
and  25? 

12.  Show  that  a  mean  proportional  between  any  two  numbers 
has  the  sign  ± . 

When  a  :  b  =  c  :  d,  prove  that  the  following   proportions   are 
true  by  deriving  them  from  a:b  =  c:  d: 

Id.    d:b  =  c:a.  -16.    a^:b^c^  =  l:d\ 

^„  b  d 

17.    ma  :  -  =  mc  :  — 

2  2 


b   a 

"  17 

.      63.^3^  ^3.^3. 

^18 

acibd  =  c^ :  d^. 
^19.    Vad  :  V6  =  Vc  :  1.  ^ 

20.    a  +  b:c  -{-  d  =  a  —  b:c  —  d, 

21.  a^  4-  a'^ft  ■j-ab'-{-b^:a^  =  €^-\-  c^d  +  cd^  -j-  c/^ :  c». 

22.  2a  +  3c:2a-3c  =  86  +  12d:86-12d. 

23.  Solve  the  equation  ^^^^t^  =  ^. 

^  ox         20 


Solution 

ax         20* 

Dividing  by  2,  ^T^  =  I^- 

2  ax        40 

Regarding  this  equation  as  a  proportion, 

,■,...        a2  +  2ax  +  x2     81 
by  composition  and  division,  ^^—^ — i±—L —  =  _. 
^         ^  '  a2  -  2  ax  +  x2      1 

a  4-  X      9 
Extracting  the  square  root,  — 


By  composition  and  division, 


a 

-  X 

1 

2a_ 
2x 

10 
"  8' 

.'.   X- 

=  |a. 

336  _  ACADEMIC  ALGEBRA 


24.   Solve  the  equation  ^'^  +  ^  +  ^'^  ^  V»  +  7  -  V^. 

4  H-  Vic  4  —  Va; 


Solution 


Va;+7  +V^_  Va;4-7  -Va; 
4  +  VS  4  —  Vx 


By  alternation,  Prin.  4,     ^^'^+^+^  =  ^+^^. 
v^x  +  7  -  Vx     4:-Vx 

By  composition  and  division,  Prin.  8, 


2Vx+7  ^    8 
2v^         2Vx 
Since  the  consequents  are  equal,  the  antecedents  are  equal. 
Therefore,  2  Va;  +  7  =  8. 

Whence,  reducing,  oj  =  9. 


25.    Given  V^  +  11  +  2^  Vg^+U  +  2|^  ^^  ^^^  ^ 
Va;  +  11  -  2     V2a;  +  14-2| 

Solution 

Va;  4-  1 1  +  2  _  V2a;  +  14  +  2f_ 
VxTn  -  2      V2X+14  -  2f 
By  composition  and  division,  Prin.  8, 


2Va;+  ll_2V2a;  +  14 
•  4  1/ 

Dividing  both  terms  of  each  ratio  by  2,  Prin.  10, 

Va;  +  11_  V2a;  +  14 
2  I 

Dividing  the  consequents  by  |,  Prin.  10, 


Vx  +  ll_  V2a;+14 
3  4 


By  alternation,  Prin.  4,  y/x-\-U    ^ 3 

V2a;+14     4 

Squaring,  and  applying  Prin.  7,  ^-i-H  =  -. 

x  +  3      7 

Solving,  X  =  26. 


RATIO  AND  PROPORTION  337 


Solve  by  the  principles  of  proportion 


26.    V^  +  V'»  =  !^.  29. 

Va;  —  Vm      ^ 

27.  vi+v||^2.  3„ 

V'a!-V2a     1 


Vx 

4-5_l_Vx- 

-b_ 

=  ct 

Vx 

^'x- 

_1 

a 

ax  - 

Va 

—  X 

Va  — Va 
■\/ax  —  b 

—  iC 

3V 

-25 

28.  £+y|^=i3.  31. 

a;  —  Vic  —  1       7  Vcix  +  h      3  V«a;  +  56 

32.  Given  ^^^±^^±1  =  ?,  to  find  X. 

ViC  +  2  +  Va;  —  1      1 

33.  Given  ^  +  ^^^+^  ^  ^^  +  ^^^.  to  find  c.. 

Va  —  Va  +  a;      V6  —  Va;  —  6 


-.     /-»•         a  —  V2  ax  —  Qc^     a  —  h   .    £■   a 
34.    Given = ,  to  find  x. 

a-\-^2ax-x^     a  +  ^ 


35.  Given  ^^+^  +  V^^=2^  V^33  +  v^^34^  ^^  ^^^  ^ 

Va;  H-  1  —  Va;  —  2      Va;  —  3  —  Va;  —  4 

36.  Divide  $  35  between  two  men  so  that  their  shares  shall  be 
in  the  ratio  of  3  to  4. 

37.  Two  numbers  are  in  the  ratio  of  3  to  2,  and  if  each  is 
increased  by  4,  the  sums  will  be  in  the  ratio  of  4  to  3.  What  are 
the  numbers  ?        \  - 

38.  Divide  16  into  two  parts  such  that  their  product  is  to  the 
sum  of  their  squares  as  3  is  to  10.         \  o    <-  ^^  ' ' 

39.  Divide  25  into  two  parts  such  that  the  greater  increased 
by  1  is  to  the  less  decreased  by  1  as  4  is  to  1.    ,  ^  ^^--4.0 

40.  The  sum  of  two  numbers  is  4,  and  the  square  of  their  sum 
is  to  the  sum  of  their  squares  as  8  is  to  5.     What  are  the  numbers  ? 

41.  A  dealer  had  two  casks  of  wine.  From  the  larger  he  drew 
34  gallons,  and  from  the  smaller  8  gallons,  after  which  their  con- 
tents were  as  5  to  4.  When  half  the  original  contents  of  each 
cask  had  been  drawn,  he  put  8  gallons  into  the  larger  and  6  into 
the  smaller.  If  the  ratio  of  their  contents  was  then  5  to  3,  what 
was  the  capacity  of  each?  q\'    . 

ACAD.   ALG.  — 22 


VARIATION 


343.  One  quantity  or  number  is  said  to  vary  directly  as  another, 
or  simply  to  vary  as  another,  when  they  depend  on  eacli  other  in 
such  a  manner  that  if  one  is  changed  the  other  is  changed  m  the 
same  ratio. 

Thus,  if  a  man  earns  a  certain  sum  per  day,  the  amount  of  wages  he  earns 
varies  as  the  number  of  days  he  works. 

344.  The  Sign  of  Variation  is  oc.     It  is  read  '  varies  as.' 

345.  The  expression  xcx:y  means  that  if  x  is  doubled,  y  is 
doubled,  or  if  x  is  divided  by  a  number,  y  is  divided  by  the  same 
number,  etc. ;  that  is,  that  the  ratio  of  a?  to  2/  is  always  the  same, 
or  constant. 

If  the  constant  ratio   is  represented  by  Jc,  then  when  xccy, 

-=k,  or  x  =  ky.     Hence, 

If  x  varies  as  y,  x  is  equal  to  y  multiplied  by  a  constant 

346.  One  quantity  or  number  varies  inversely  as  another  when 
it  varies  as  the  reciprocal  of  the  other. 

Thus,  the  time  required  to  do  a  certain  piece  of  work  varies  inversely  as 
the  number  of  men  employed.  For,  if  it  takes  10  men  4  days  to  do  a  piece  of 
work,  it  will  take  5  men  8  days,  or  4  men  10  days,  to  do  it. 

1  1  X 

In  a;  cc  -,  if  the  constant  ratio  of  «  to  -  is  A:,  :j-  =  A:,  oi  xy  =  k. 
y  2/1 

Hence,  y 

If  X  varies  inversely  as  y,  their  product  is  constant. 

347.  One  quantity  or  number  varies  jointly  as  two  others  when 
it  varies  as  their  product. 

338 


VARIATION  339 

Thus,  the  amount  of  money  a  man  earns  varies  jointly  as  the  number  of 
days  lie  works  and  the  sum  he  receives  per  day.  For,  if  he  should  work 
three  times  as  many  days,  and  receive  twice  as  many  dollars  per  day,  he 
would  receive  six  times  as  much  money. 

In  a:  oc  yz,  if  the  constant  ratio  of  x  to  yz  is  h, 

—  =  k,  or  X  =  kyz.     Hence, 
yz 

If  X  varies  jointly  as  y  and  z,  x  is  equal  to  their  product  multiplied 
by  a  constant. 

348.  One  quantity  or  number  varies  directly  as  a  second  and 
inversely  as  a  third  when  it  varies  jointly  as  the  second  and  the 
reciprocal  of  the  third. 

Thus,  the  time  required  to  dig  a  ditch  varies  directly  as  the  length  of  the 
ditch  and  inversely  as  the  number  of  men  employed.  For,  if  the  ditch  were 
10  times  as  long  and  5  times  as  many  men  were  employed,  it  would  take 
twice  as  long  to  dig  it. 

1  ?/ 

Inxccy  '  -i  01  xcc  -,  if  A;  is  the  constant  ratio, 
z  z 

x-7--  =  k,  or  x  =  k '-'     Hence, 

2       '  z  ' 

If  X  varies  directly  as  y  and  inversely  as  z,  x  is  equal  to  -1  multi- 
plied by  a  constant. 

349.  If  X  varies  as  y  when  z  is  constant,  and  x  varies  as  z  when 
y  is  constantj  then  x  vanes  as  yz  when  both  y  and  z  are  variable. 

Proof.  —  Since  the  variation  of  x  depends  on  the  variations  of  y  and  ^, 
suppose  the  latter  variations  to  take  place  in  succession,  each  in  turn  pro- 
ducing a  corresponding  variation  in  x. 

While  z  remains  constant,  let  y  change  to  yi*  thus  causing  x  to  change 
to  x'. 

Then,  '^=y-.  (1) 

^'    yi 

Now  while  y  keeps  the  value  ?/i,  let  z  change  to  ^i,  thus  causing  x'  to 
change  to  Xi. 

Then,  ^  =  £.  (2) 

Xi       Zi 

*  In  algebraic  notation  Xi,  Xg,  xa,  etc.,  are  read  'x  sub  one,'  'x  sub  two,' 
'X  sub  three,'  etc. 


340  ACADEMIC  ALGEBRA 


Multiplying  (1)  by  (2), 

X       yz 
xi     yizi 

X\ 

.:  X  = yz  = 

=  Tcyz, 

where  k  is  the  constant  ^^• 
yi^i 
Hence, 

X  (x:yz. 

Thus,  the  area  of  a  triangle  varies  as  the  base  when  the  altitude  is  con- 
stant, varies  as  the  altitude  when  the  base  is  constant,  and  varies  as  the 
product  of  the  base  and  altitude  when  both  vary. 

Similarly,  if  x  varies  as  each,  of  three  or  more  numbers,  y,  z, 
V,  '•'  when  the  others  are  constant,  when  all  vary  x  varies  as  their 
product. 

That  is,  xccyzv*". 

Thus,  the  volume  of  a  parallelepiped  varies  as  the  length,  if  the  width  and 
thickness  are  constant ;  as  the  width,  if  the  length  and  thickness  are  con- 
stant ;  as  the  thickness,  if  the  length  and  width  are  constant ;  as  the  product 
of  any  two  dimensions,  if  the  other  is  constant;  or  as  the  product  of  the 
three  dimensions,  if  all  vary. 

\^-  Examples 

350.  1.  If  ic  varies  inversely  as  y,  and  x  =  6  when  y  =  S,  what 
is  the  value  of  x  when  y  =  12? 

Solution 

Since  a;  x  -,  let  k  be  the  constant  ratio  of  x  to  — 
y  y 

Then,  §  346,  xy  =  k.  (1) 

Hence,  when  ic  =  6  and  ?/  =  8, 

A;  =  6  X  8,  or  48.  (2) 

Since  k  is  constant,  k  =  4:S  when  y  =  12. 

Hence,  Eq.  (1)  becomes  12ic  =  48. 

Therefore,  when  y  =  12,  x  =  4. 

2.  The  volume  of  a  cone  varies  jointly  as  its  altitude  and  the 
square  of  the  diameter  of  its  base.  When  the  altitude  is  15  and 
the  diameter  of  the  base  is  10,  the  volume  is  392.7.  What  is  the 
volume,  when  the  altitude  is  5  and  the  diameter  of  the  base 
is  20  ? 


VARIATION  341 


Solution 


Since 

VccHD^  or  V  =  kHD^, 

d 

V  =  392.7  when  H=  15  and  D  =  10, 

392.7  =  A;  X  15  X  100. 

Also,  since 

V  becomes  v  when  H=d  and  D  =  20, 

^  =  A;  X  5  X  400. 

Dividing  (2)  by  (1).      ,^_,  =  ^\IZ  =  I- 

Let  F,  -ff,  and  D  denote  the  volume,  altitude,  and  diameter  of  the  base, 
respectively,  of  any  cone,  and  v  the  volume  of  a  cone  whose  altitude  is 
5  and  the  diameter  of  whose  base  is  20. 


(1) 


(2) 
(3) 

.-.  1?  =  ^  of  392.7  =  523.6. 
3.   If  xccy  and  yocz,  prove  that  xocz. 

Proof 

Since  xccy  and  y<KZ,  let  m  represent  the  constant  ratio  of  x  to  y,  and 
n  the  constant  ratio  of  y  to  z. 

Then,  §  346,  x  =  my,  (1) 

And  y  =  nz.  (2) 

Substituting  n^  for  y  in  (1),       jc  =  wn2!.  (3) 

Hence,  since  mn  is  constant,  a;  a  2. 

V  4.  The  circumference  of  a  circle  varies  as  its  diameter.  If 
the  circumference  of  a  circle  whose  diameter  is  1  foot  is  3.1416 
feet,  what  is  the  circumference  of  a  circle  whose  diameter  is 
100  feet? 

i/'  5.  The  area  of  a  circle  varies  as  the  square  of  its  diameter. 
If  the  area  of  a  circle  whose  diameter  is  10  feet  is  78.54  square 
feet,  what  is  the  area  of  a  circle  whose  diameter  is  20  feet  ? 

^'6.  The  distance  a  body  falls  from  rest  varies  as  the  square 
of  the  time  of  falling.  If  a  stone  falls  64.32  feet  in  2  seconds, 
how  far  will  it  fall  in  5  seconds  ? 

^^  7.  The  area  of  a  triangle  varies  jointly  as  its  base  and  alti- 
tude. The  area  of  a  triangle  whose  base  is  12  inches  and  altitude 
6  inches  is  36  square  inches.  What  is  the  area  of  a  triangle 
whose  base  is  8  inches  and  altitude  10  inches?  What  is  the 
constant  ratio  ? 


342  ACADEMIC  ALGEBRA 

8.  A  wrought  iron  bar  1  square  inch  in  cross  section  and 
1  yard  long  weighs  10  pounds.  If  the  weight  of  a  uniform  bar 
of  given  material  varies  jointly  as  its  length  and  the  area  of  its 
cross  section,  what  is  the  weight  of  a  wrought  iron  bar  3(5  feet 
long,  4  inches  wide,  and  4  inches  thick  ? 

^  9.  The  weight  of  a  beam  varies  jointly  as  the  length,  the  area 
of  the  cross  section,  and  the  material  of  which  it  is  composed. 
If  wood  is  yL  as  heavy  as  wrought  iron,  what  is  the  weight  of  a 
wooden  beam  24  feet  long,  12  inches  wide,  and  12  inches  thick  ? 

-4-  10.  What  is  the  weight  of  a  brick  2  in.  x  4  in.  x  8  in.,  if  the 
material  is  \  as  heavy  as  wrought  iron  ? 

Yll.  If  10  men  can  do  a  piece  of  work  in  20  days,  how  long  will 
it  take  25  men  to  do  it  ? 

i  12.  If  a  men  can  do  a  piece  of  work  in  b  days,  how  many  men 
will  be  required  to  do  it  in  c  days  ? 

/-IS.  The  distances,  from  the  fulcrum  of  a  lever,  of  two  weights 
that  balance  each  other  vary  inversely  as  the  weights.  If  two 
boys  weighing  80  and  90  pounds,  respectively,  are  balanced  on  the 
ends  of  a  board  8^  feet  long,  how  much  of  the  board  has  each  ? 

/'14.  A  water  carrier  carries  two  buckets  of  water  suspended 
from  the  ends  of  a  4-foot  stick  that  rests  on  his  shoulder.  If 
one  bucket  weighs  60  pounds  and  the  other  100  pounds,  and  they 
balance  each  other,  what  point  of  the  stick  rests  on  his  shoulder  ? 

/ 15.  The  weight  of  a  body  near  the  earth  varies  inversely  as 
the  square  of  its  distance  from  the  center  of  the  earth.  If  the 
radius  of  the  earth  is  4000  miles,  what  would  be  the  weight  of  a 
4-lb.  brick  at  the  distance  of  4000  miles  from  the  earth's  surface  ? 

16.  A  boy  wishes  to  ascertain  the  height  of  a  tower.  He 
knows  that  it  is  31  feet  6  inches  from  his  window  to  the  pave- 
ment below,  and  that  the  distance  through  which  a  body  falls 
varies  as  the  square  of  the  time  of  falling.  He  drops  a  marble 
from  his  window  and  finds  that  it  strikes  the  pavement  in  1.4 
seconds.  Then  he  throws  a  stone  to  the  top  of  the  tower  and 
observes  that  it  takes  just  3  seconds  for  it  to  descend.  What 
is  the  height  of  the  tower  ? 


VARIA  TION  343 

17.  A  horse  tied  with  a  rope  45  feet  long  to  a  stake  m  the 
center  of  a  pasture  eats  all  the  grass  within  reach  in  li  days.  If 
his  rope  were  15  feet  longer,  how  many  days  would  it  take  him  to 
eat  all  the  grass  within  reach  ? 

18.  The  illumination  from  a  source  of  light  varies  inversely 
0      as  the  square  of  the  distance.     How  far  must  a  screen  that  is  10 

feet  from  a  lantern  be  moved  so  as  to  receive  one  fourth  as  much 
light  ? 

*>^  19.    The  number  of  times  a  pendulum   oscillates  in  a  given 
^^  time  varies  inversely  as  the  square  root  of  its  length.     If  a  pen- 
^'       dulum  39.1   inches  long  oscillates  once  a  second,  what  is  the 
length  of  a  pendulum  that  oscillates  twice  a  second? 

^^  ^20.    How  long  must  a  pendulum  be  to  oscillate  once  in  three 
^P^     seconds  ? 

21.  If  xx -,  and  if  a;  =  2  when  y  =  12  and  z  =  2,  what  is  the 
value  of  X  when  y  =  84:  and  z  =  7  ? 

22.  If  ojoc-,  and  if  a;  =  60  when  y  =  24  and  z  =  2y  what  is  the 
value  of  y  when  a;  =  20  and  z  =  7  ? 

23.  If  X  varies  jointly  as  y  and  z  and  inversely  as  the  square 
of  Wj  and  if  a;  =  30  when  y  =  S,  z  =  5,  and  w;  =  4,  what  is  the 
value  of  X  expressed  in  terms  of  y,  z,  and  iv? 

24.  If  xcc-  and  voc-,  prove  that  xccz. 

y  ^     z'  ^ 

25.  li  xccy  and  zcx^y,  prove  that  (x  ±  z)ccy. 

-^\y  26.  Three  spheres  of  lead  whose  radii  are  6,  8,  and  10  in.,  re- 
spectively, are  united  into  one.  Find  the  radius  of  the  resulting 
sphere,  if  the  volume  of  a  sphere  varies  as  the  cube  of  its  radius. 

^^_,  27.  The  volume  of  a  cone  varies  jointly  as  its  altitude  and  the 
square  of  the  diameter  of  its  base.  The  altitudes  of  three  cones, 
S,  P,  and  R,  are  30  ft.,  10  ft.,  and  5  ft.,  respectively.  The 
diameter  of  the  base  of  P  is  5  ft.  and  that -of  11  is  10  ft.  If  the 
volume  of  H  is  equivalent  to  that  of  P  and  R  combined,  what  is 
the  diameter  of  the  base  of  /S? 


PROGRESSIONS 


351.  1.  How  does  each  of  the  numbers  2,  4,  6,  8,  10,  12,  ••• 

compare  with  the  number  that  follows  it  ?     How  may  any  term 
after  the  first  be  obtained  from  the  preceding  term  ? 

2.  How  may  any  term  of  2,  2^,  3,  3 J,  •••  after  the  first  be  ob- 
tained from  the  preceding  term  ? 

3.  Write  a  series  of  six  terms  beginning  with  a  and  increasing 
by  a  constant  number  d. 

4.  How  may  any  term,  after  the  first,  of  the  series  3,  6,  12, 
24,  48,  •••  be  obtained  from  the  preceding  term? 

5.  How  may  any  term,  after  the  first,  of  the  series  1,  i,  i,  i?  ••• 
be  obtained  from  the  preceding  term  ? 

6.  Write  a  series  of  six  terms  beginning  with  a  and  increasing 
by  a  constant  multiplier  r. 

352.  A  succession  of  numbers,  each  of  which  after  the  first  is 
derived  from  the  preceding  number  or  numbers  according  to  some 
fixed  law,  is  called  a  Series. 

353.  The  successive  numbers  are  called  the  Terms  of  the  series. 
The  first  and  last  terms  of  a  series  are  the  Extremes,  the  inter- 
vening terms  the  Means. 

In  the  series  a,  a  +  d,  a  +  2  (?,  a  +  3  <^,  a  -t-  4  (^,  the  terms  a  and  a  -h  4  d 
are  the  extremes  and  the  other  terms  are  the  means. 

354.  A  series  consisting  of  a  limited  number  of  terms  is 
called  a  Finite  Series. 

355.  A  series  consisting  of  an  unlimited  number  of  terms  is 
called  an  Infinite  Series. 

344 


/ 


PROGRESSIONS  345 

ARITHMETICAL  PROGRESSION 

356.  A  series  each  term  of  which  after  the  first  is  derived  from 
the  preceding  by  the  addition  of  a  constant  number  is  called  an 
Arithmetical  Series,  or  an  Arithmetical  Progression. 

357.  The  number  added  to  any  term  to  produce  the  next  is 
called  the  Common  Difference. 

2,  4,  6,  8,  •••  and  16,  12,  9,  6,  •••  are  arithmetical  progressions.  In  the 
first,  the  common  difference  is  2  and  the  series  is  ascending ;  in  the  second, 
the  common  difference  is  —  3  and  the  series  is  descending. 

A.  P.  is  an  abbreviation  of  the  words  Arithmetical  Progression. 

358.  To  find  the  nth,  or  last  term. 

1.  In  the  arithmetical  progression  a;,  a;  +  2,  a;  +  4,  a;  +  6,  what 
is  the  common  difference  ?  How  many  times  does  it  enter  into 
the  second  term  ?  into  the  third  term  ?  into  the  fourth  term  ? 

2.  From  the  first  term  of  the  series  a,  a  -f-  d,  a  +  2  d,  a  +  3  d,  •  •  • 
how  is  the  second  term  formed  ?  the  third  term  ?  the  fourth 
term  ?  the  fifth  term  ?  the  nth  term,  or  any  term  ? 

3.  What  is  the  nth  term  of  the  series  a,  a  —  d,  a  —  2  c?,  •••  ? 


359.   When  a  represents  the  first  term  of  an  A.P.,  d  the  com- 
mon difference,  I  the  nth,  or  last  term,  and  n  the  number  of  terms, 

Z  =  a  +  (?i-l)d  (I) 


Examples 

1.   What  is  the  10th  term  of  the  series  3,  6,  9,  ••.  ? 

PROCESS  Explanation.  —  Since  the  series  3,  6,  9,  ...  is  an 

l  =  a  -{-  (n  —  V)d  A. P.  the  common  difference  of  whose  terms  is  3,  sub- 

;  _  3  _^  no  —1)3  stituting  3  for  a,  3  for  d,  and  10  for  n  in  the  formula 

7 oA  for  the  last  term,  the  last  term  is  found  to  be  30. 

^^2.   Find  the  20th  term  of  the  series  7,  11,  15,  •••. 

3.  Find  the  16th  term  of  the  series  2,  7,  12,  ••.. 

4.  Find  the  24th  term  of  the  series  1,  16,  31,  •••. 


346  ACADEMIC  ALGEBRA 

6.    Find  the  IStli  term  of  the  series  1,  8,  15,  •••. 

6.  Find  the  13th  term  of  the  series  —  3,  1,  5,  •••. 

7.  Find  the  49th  term  of  the  series  1,  li  If,  •••- 

8.  Find  the  15th  term  of  the  series  45,  43,  41,  ••.. 
Suggestion. — The  common  difference  is  —  2. 

9.  Find  the  10th  term  of  the  series  5,  1,  —  3,  •••. 

10.  Find  the  16th  term  of  the  series  a,  3  a,  5  a,  •••. 

11.  Find  the  12th  term  of  the  series  a  —  b,  a -\-b,  a -{- Sb.  •••„ 

12.  Find  the  7th  term  of  the  series  x  —  Sy,x  —  2y,  •••. 

13.  A  body  falls  16^  feet  the  first  second,  3  times  as  far  the 
second  second,  5  times  as  far  the  third  second,  etc.     How  far  will 

\     it  fall  during  the  lO'th  second  ? 

\ 

360.  To  find  the  sum  of  n  terms  of  a  series. 

1.  Express  5  terms  of  the  series  a,  a -}-  d,  a -\-2  d,  '•-. 

2.  How  may  the  term  before  the  last  term  be  obtained  from 
the  last  term  ?  If  I  represents  the  last  term  and  d  the  common 
difference,  what  will  be  the  term  next  to  the  last  ?  the  second 
term  from  the  last  ?  the  third  term  from  the  last  ? 

3.  How,  then,  may  the  series  a,  a  +  d,  •••  be  written  in  reverse 
order,  if  the  last  term  is  I  ? 

361.  Let  a  represent  the  first  term  of  an  A.P.,  d  the  common 
difference,  I  the  last  term,  n  the  number  of  terms,  and  s  the  sum 
of  the  terms. 

Writing  the  sum  of  n  terms  in  the  usual  order  and  then  in  the 
reverse  order,  and  adding  the  two  equal  series, 

s  =  a  +  («  +  (^)  +  (a  +  2  d)  -f-  (a  +  3  d)  H \-l 

s  =  I  +  (I  -  d)  -^  (I  -  2  d)  -j-  (I  -  3  d)  -\-  •"  -^  a. 

2  5  =  (a  -h  0  +  («  +  0  +  («  +  0  +  («^  +  0  +  •••  +  («  +  0- 
.-.  2  s  =  n  (a  +  I). 

.=  |(a  +  0,orn(^Mli^.  (II) 


PROGRESSIONS  347 


Examples 
1.    What  is  the  sum  of  20  terms  of  the  series  2,  5,  8,  •••  ? 

PROCESS 

Z  =  a  +  (71  -  1)  (^  =  2  +  (20  -  1^  X  3  =  59 


s  =  n(^y20(^-^\=m 


y- 


Explanation.  —  Since  the  last  term  is  not  given,  it  is  found  by  the  pre- 
vious case  and  substituted  for  I  in  the  formula  for  the  sum. 

'    2.  What  is  the  sum  of  16  terms  of  the  series  1,  5,  9,  •••  ? 

3.  What  is  the  sum  of  10  terms  of  the  series  —  2,  0,  2,  •••  ? 

4.  What  is  the  sum  of  6  terms  of  the  series  1,  3i,  6,  •••  ? 

5.  What  is  the  sum  of  8  terms  of  the  series  a,  3  a,  5  «,  •••  ? 

6.  What  is  the  sum  of  n  terms  of  the  series  1,  7,  13,  •••  ? 

7.  What  is  the  sum  of  a  terms  of  the  series  x^  x-{-2a,  •••  ? 

8.  What  is  the  sum  of  7  terms  of  the  series  4,  11,  18,  •••  ? 

9.  What  is  the  sum  of  10  terms  of  the  series  1,  —  1,  —  3,  •••  ? 

10.  What  is  the  sum  of  10  terms  of  the  series  1,  i,  0,  •••  ? 

11.  How  many  times  does  a  common  clock  strike  in  12  hours  ? 

12.  A  body  falls  lOy^^  feet  the  first  second,  3  times  as  far  the 
second  second,  5  times  as  far  the  third  second,  etc.  How  far  will 
it  fall  in  10  seconds  ? 

13.  Thirty  flower  pots  are  arranged  in  a  straight  line  4  feet 
jj>     apart.     How  far   must  a  lady  walk  who,  after  watering  each 

plant,  returns  to  a  well  4  feet  from  the  first  plant  and  in  line 
with  the  plants,  assuming  that  she  starts  at  the  well  ? 

14.  A  boy  took  a  30-day  job  on  the  following  terms :  he  was 
^  to  receive  5  cents  the  first  day,  10  cents  the  second  day,  15  cents 
^  /    the  third  day,  etc.     How  much  was  he  paid  for  the  thirtieth  day, 

and  what  was  the  whole  amount  of  his  earnings  ? 


348  ACADEMIC  ALGEBRA 

362.   The  two  fundamental  formulae, 


n 


(I)  l  =  a  +  (n-l)d  and  (II)  s=^(a-\-l), 

contain  jive  elements^  a,  d,  I,  n,  and  s.  Since  these  formulae  are 
independent  simultaneous  equations,  if  they  contain  but  two 
unknown  elements  they  may  be  solved.  Hence,  if  any  three  of 
the  five  elements  are  known,  the  other  two  may  be  found. 

Examples 

1.  The  last  term  of  an  A.  P.  is  58,  the  common  difference  is  3, 
and  the  sum  of  the  series  is  260.  Find  the  number  of  terms  and 
the  first  term. 

Solution 

Substituting  58  for  Z,  3  for  d,  and  260  for  s  in  both  (I)  and  (II) 


(I)  becomes 

58  =  a  +  (w  -  1)3. 

(1) 

(II)  becomes 

260  =  ^  (a +  58). 

(2) 

(1)  X  n, 

58  n  =  wa  +  3  n2  -  3  n. 

(3) 

(2)  X  2, 

520  =  wa  +  58  n. 

(4) 

(3) -(4), 

58  w  -  520  =  3  W2  -  61  n. 

(5) 

Zrfi 

-  119  w  +  520  =  0. 

(n- 

5)(3w-104)=0. 

/.  w  =  5,  the  number  of  terms. 

Substituting  ir 

^(1), 

a  =  46,  the  first  terai. 

Since  the  number  of  terms  must  be  expressed  by  a  positive  integer,  frac- 
tional or  negative  values  of  n  are  rejected. 

2.    How  many  terms  are  there  in  the  A.  P.  2,  5,  8,  ••.,  if  the 
sum  is  610  ? 

Solution 

Since  a,  d,  and  s  are  given,  and  w,  but  not  Z,  is 'required,  n  may  be  found 
by  eliminating  I  from  (I)  and  (II)  and  solving  the  resulting  equation. 

From  (I)  and  (II),      l  =  a  +  {n -\)d  =  ^ 


n 
Substituting  2  for  a,  3  for  d,  and  610  for  s,  and  solving, 

w  =  20. 


PROGRESSIONS  349 


/ 


3.  How  many  terms  are  there  in  the  series  2,  6,  10,  •••  66  ? 

4.  What  is  the  sum  of  the  series  1,  6,  11,  •••  61  ? 

5.  How  many  terms  are  there  in  the  series  —  1,  2,  5,  •••,  if 
the  sum  is  221  ? 

6.  Determine  the  series  2,  9,  16,  •••  86. 

7.  Determine  the  series  —  10,  —  8^,  —  7,  •••  to  10  terms. 

8.  The  sum  of  the  series  .-.22,  27,  32,  •.•  is  714.     If  there 
are  17  terms,  what  are  the  first  and  last  terms  ? 

9.    If  s  =  113f ,  a  =  i,  and  d  =  2,  find  n. 

10.  What  is  the  sum  of  the  series  -  16,  —  11,  —  6,  .••  34? 

11.  What  is  the  sum  of  the  series  •••  —  1,  3,  7,  •••  23,  if  the 
number  of  terms  is  16  ? 

12.  What  are  the  extremes  of  the  series  •••  8,  10,  12,  •..,  if  5  = 
300,  and  ?i  =  20  ? 

13.  How  many  terms  are  there  in  the  series  1,  5,  9,  •••  ^? 

14.  What  is  the  sum  of  an  A.  P.  whose  extremes  are  x  and  y, 
if  the  number  of  terms  is  b  ? 

363.   To  insert  arithmetical  means. 

Examples 

1.  Insert  5  arithmetical  means  between  1  and  31. 

Solution.  —  Since  there  are  5  means,  there  must  be  7  terms.     Hence,  in 
l  =  a  -\-(n  —  l)d,  Z  =  31,  a  =  1,  n  =  l,  and  d  is  unknown. 

Solving,  d  =  5. 

Or,  since  there  are  5  means,  there  must  be  6  terms  after  the  flrst. 

tj  ^     31  -  1      .  ^ 

Hence,  a  = =  5. 

6 

.-.  1,  6,  11,  16,  21,  26,  31,  is  the  series. 

2.  Insert  9  arithmetical  means  between  1  and  6. 

3.  Insert  10  arithmetical  means  between  24  and  2. 

4.  Insert  7  arithmetical  means  between  10  and  —  14. 

5.  Insert  6  arithmetical  means  between  —  1  and  2. 


350  ACADEMIC  ALGEBRA 

6.  Insert  14  arithmetical  means  between  15  and  20. 

7.  Insert  3  arithmetical  means  between  a  —  b  and  a  -\-h. 

8.  Deduce  the  formula  for  the  common  difference  when  m 
arithmetical  means  are  to  be  inserted  between  a  and  l.  Find  the 
first  mean. 

9.  What  is  the  arithmetical  mean  between  2  and  6  ?   between 

\10  and  20  ?   between  —  3  and  5  ?    between  a  and  b  ? 
364.    Principle.  —  The  arithmetical  mean  between  two  numbers 
is  equal  to  half  their  sum. 

The  above  principle  may  be  established  as  follows : 

Let  a  and  b  represent  any  two  numbers,  and  A  their  arithmetical  mean. 

It  is  uO  be  proved  that  A  =  -^ — 

Since  the  two  numbers  and  their  arithmetical  mean  form  the  arithmetical 
progression  a,  A,  b, 

§  356,  A-a  =  b-A, 

2A  =  a  +  b. 

/.  ^  =  «^. 
2 

Examples 
Find  the  arithmetical  mean  between 

1-   S^^^i-  4.  ^^tl  and  ^^::^. 

X  —  y  X  -\-  y 

2.  a  -\-  b  and  a  —  b.  2 

5.    1  —  X  and  ^^ — ~ — ^' 

3.  (a  +  bf  and  (a  -  bf.         -  1+x 

Problems 

365.  Problems  in  Arithmetical  Progression  involving  two 
unknown  elements  commonly  suggest  series  of  the  form 

'x,   x-^y,   x-\-2y,   x  +  Sy,   etc. 

Frequently,  however,  the  solution  of  problems  is  more  readily 
accomplished  by  representing  the  series  as  follows : 

1.   When  there  are  three  terms,  the  series  may  be  written 
x-y,   X,   x  +  y. 


PROGRESSIONS  351 

2.  When  there  are  five  terms,  the  series  may  be  written, 

05  -  2  ?/,  X  -  y,  «,  X  +  y,  X  +  2  y. 

3.  When  there  are  four  terms,  the  series  may  be  written, 

X  -  3  y,  X  -  y,  X  +  y,  X  +  3  y. 

The  sum  of  the  terras  of  a  series  represented  as  above  evidently  contains 
but  one  unknown  number. 

1.  The  sum  of  three  numbers  in  arithmetical  progression  is 
30,  and  the  sum  of  their  squares  is  462.     What  are  the  numbers  ? 

Solution 
Let  the  series  be  x  —  y,  x,  x  +  y. 

Then,  (x  -  y)+ x +  (x  +  y)=  30,  (1) 

and  (x  -  y)2  +  x2  +  (x  +  yf  =  462.  (2) 

From  (1),  3x  =  30.  (3) 

X  =  10.  (4) 

From  (2 ) ,  8  x2  +  2  y2  =  462.  (5) 

Substituting  10  for  x,  2  y2  =  i62.  (6) 

Solving,  y  =  ±  9. 

Forming  the  series  from  x  =  10  and  y  =  ±  9,  the  terms  are 
1,  10,  19,  or  19,  10,  1. 

2.  The  sum  of  three  numbers  in  arithmetical  progression  is 
18,  and  their  product  is  120.     What  are  the  numbers  ? 

3.  The  sum  of  three  numbers  in  arithmetical  progression  is 
21,  and  the  sum  of  their  squares  is  155.     What  are  the  numbers  ? 

4.  There  are  three  numbers  in  arithmetical  progression  the 
sum  of  whose  squares  is  93.  If  the  third  is  4  times  as  large  as 
the  first,  what  are  the  numbers  ? 

5.  The  product  of  the  extremes  of  an  arithmetical  progression 
of  3  terms  is  4  less  than  the  square  of  the  mean.  What  are 
the  numbers,  if  their  sum  is  24  ? 

6.  The  sum  of  four  numbers  in  arithmetical  progression  is  14, 
and  the  product  of  the  means  is  12.     What  are  the  numbers  ? 

7.  The  sum  of  seven  numbers  in  arithmetical  progression  is 
98,  and  the  sum  of  their  squares  is  1484.    What  are  the  numbers '/ 


-\ 


S52  ACADEMIC  ALGEBRA 

8.  The  sum  of  five  numbers  in  arithmetical  progression  is  15, 
and  the  product  of  the  extremes  is  3  less  than  the  product  of  the 
terms  next  to  the  extremes.     What  are  the  numbers  ? 

9.  A  number  is  expressed  by  three  digits  in  arithmetical  pro- 
gression. If  the  number  is  divided  by  the  sum  of  its  digits,  the 
quotient  is  20^ ;  and  if  the  number  is  increased  by  594,  the  result 
is  the  number  with  its  digits  in  the  reverse  order.  What  is 
the  number? 

10.  Find  the  sum  of  the  odd  numbers  from  1  to  100. 

11.  The  product  of  the  extremes  of  an  arithmetical  progression 
of  10  terms  is  70,  and  the  sum  of  the  series  is  95.  What  are  the 
extremes  ? 

12.  Fifty-five  logs  are  to  be  piled  so  that  the  top  layer  shall 
consist  of  1  log,  the  next  layer  of  2  logs,  the  next  layer  of  3  logs, 
etc.     How  many  logs  must  be  placed  in  the  bottom  layer  ? 

13.  It  cost  Mr.  Smith  $  19.00  to  have  a  well  dug.  If  the  cost 
of  digging  was  $1.50  for  the  first  yard,  $1.75  for  the  second, 
$  2.00  for  the  third,  etc.,  how  deep  was  the  well  ? 

14.  The  product  of  the  extremes  of  an  arithmetical  progression 
of  15  terms  is  93,  and  the  sum  of  the  first  and  last  means  is  34. 
What  is  the  progression  ? 

15.  How  many  arithmetical  means  must  be  inserted  between 
5  and  37,  so  that  the  ratio  of  the  first  mean  to  the  last  mean 
may  be  y\  ? 

16.  How  many  arithmetical  means  must  be  inserted  between 
4  and  25,  so  that  the  sum  of  the  series  may  be  116  ? 

17.  Prove  that  the  equimultiples  of  the  terms  of  an  arith- 
metical progression  are  in  arithmetical  progression. 

18.  Prove  that  the  difference  of  the  squares  of  consecutive 
integers  are  in  arithmetical  progression,  and  that  the  common 
difference  is  2. 

19.  Prove  that  the  sum  of  n  consecutive  odd  integers,  beginning 
with  1,  is  n^. 


PROGRESSIONS  353 


GEOMETRICAL  PROGRESSION 

366.  A  series  of  numbers  each  of  which  after  the  first  is  derived 
by  multiplying  the  preceding  number  by  some  constant  multiplier 
is  called  a  Geometrical  Series,  or  a  Geometrical  Progression. 

2,  4,  8,  10,  o2  and  a*,  n'^,  a^,  a  are  geometrical  progressions. 

In  the  first  series  the  constant  multiplier  is  2  ;  in  the  second  it  is  — 

G.  P.  is  an  abbreviation  of  the  words  Geometrical  Progression. 

367.  The  constant  multiplier  is  called  the  Ratio. 

It  is  evident  that  the  terms  of  a  geometrical  progression 
increase  or  decrease  numerically  according  as  the  ratio  is  numer- 
ically  greater  or  less  than  1. 

368.  To  find  the  /7th,  or  last  term. 

1.  In  the  geometrical  progression  3,  6,  12,  24,  what  is  the 
ratio  of  6  to  3  ?  of  12  to  6  ?  of  24  to  12  ? 

2.  In  the  geometrical  progression  a,  ar,  ar^,  ar^  •••  what  is  the 
ratio  ?  How  many  times  does  the  ratio  enter  as  a  factor  into  the 
second  term  ?  into  the  third  term  ?  into  the  fourth  term  ? 

369.  When  a  represents  the  first  term  of  a  G.  P.,  r  the  ratio, 
and  I  the  last  or  nth  term, 

I  =  ar^-\  (I) 


Examples 
1.    Find  the  9th  term  of  the  series  1,  3,  9,  •••. 

PROCESS 

Explanation.  — In  this  example  a  =  1,  r  =  3,  and 
I  =  ar""-^  n  =  9. 

I  =:±  X  3*  Substituting  these  values  in  the  formula  for  I,  the 

7       /.(r/.^  last  term  is  6561. 

I  —  6561 

2.  Find  the  10th  term  of  the  series  1,  2,  4,  •••. 

3.  Find  the  8th  term  of  the  series  \,  |,  1,  •••. 

4.  Find  the  9th  term  of  the  series  6,  12,  24,  .... 

ACAD.   ALG.  — 23 


354.  ACADEMIC  ALGEBRA 

5.  Find  the  11th  term  of  the  series  ^,  1,  2,  .... 

6.  Find  the  7th  term  of  the  series  2,  6,  18,  .••. 

7.  Find  the  6th  term  of  the  series  4,  20,  100,  .... 

8.  Find  the  6th  term  of  the  series  6,  18,  54,  ... 

9.  Find  the  10th  term  of  the  series  1,  ^,  ^,    •., 

10.  Find  the  10th  term  of  the  series  1,  |,  |,  •••. 

11.  Find  the  8th  term  of  the  series  J,  i,  |,  •••• 

12.  Find  the  11th  term  of  the  series  a%  a'^6V- 

13.  Find  the  nth  term  of  the  series  2,  V2,  1,  •••. 

14.  A  man  worked  for  25  cents  the  first  day,  50  cents  the 
second  day,  $1  the  third  day,  and  so  on  for  10  days.  How  much 
did  he  receive  the  tenth  day  ? 

15.  If  a  man  begins  business  with  a  capital  of  $200  and 
doubles  it  every  year  for  6  years,  how  much  will  he  have  at 
the  end  of  the  sixth  year? 

16.  If  the  population  of  the  United  States  is  76  millions  in 
1900  and  doubles  itself  every  25  years,  what  will  it  be  in  the 
year  2000  ? 

17.  A  man's  salary  was  raised  |  every  year  for  5  years.  If 
his  salary  was  $  512  the  first  year,  what  was  it  the  sixth  year  ? 

18.  The  population  of  a  city  at  a  certain  time  was  20,736,  and 
increased  in  geometrical  progression  25%  each  decade.  What 
was  the  population  at  the  end  of  40  years  ? 

19.  A  man  who  wanted  10  bushels  of  wheat  thought  $1  a 
bushel  too  high  a  price.  But  he  agreed  to  pay  2  cents  for  the 
first  bushel,  6  cents  for  the  second,  18  cents  for  the  third,  and 
so  on.     What  did  the  last  bushel  cost  him  ? 

20.  From  a  grain  of  corn  there  grew  a  stalk  that  produced 
an  ear  of  150  grains.  These  grains  were  planted,  and  each  pro- 
duced an  ear  of  150  grains.  This  process  was  repeated  until 
there  were  4  harvestings.  If  75  ears  of  corn  make  1  bushel,  how 
many  bushels  were  there  the  fourth  year  ? 


PROGRESSIONS  355 

370.    To  find  the  sum  of  a  finite  series. 

Let  a  represent  the  first  term,  I  the  nth.  term,  or  the  last  term, 
r  the  ratio,  n  the  number  of  terms,  and  s  the  sum  of  the  terms. 

Then,  s  =  a -{- ar -\- ar^  A- a)^ -\ h  ar^'-K  (1) 

(1)  X  r,  rs  =  ar -\- m^ -^  a)^ -\ h  ar""^  -f  ar\  (2) 

(2)-(l),      s(r-l)=ar"-a. 

3^^^"-^^.o,  «(^--/).  (II) 

r  —  1  r  —  1 

But,  since  ar"~^  =  I,  ar'^  =  rZ, 
Substituting  rl  for  a?-^  in  (II), 

rl-a^^a-H^  (III) 


»  = 


-1'         1-r 


Examples 
1.   Find  the  sum  of  6  terms  of  the  series  3,  9,  27,  •••. 


PROCESS 

^^ ^  Explanation.  —  Since  the  first  term  a,  the 

ratio  r,  and  the  number  of  terms  n,  are  given, 
and  formula  II  gives  the  sum  in  terms  of  a,  r. 


r-1 


^  _  3x3  —  3  _  -|^Qg2      and  n,  formula  II  is  used. 
3-1 

2.  Find  the  sum  of  8  terms  of  the  series  1,  2,  4,  •••. 

3.  Find  the  sum  of  8  terms  of  the  series  1,  |,  J,  •••. 

4.  Find  the  sum  of  10  terms  of  the  series  1,  1^,  2\,  •••. 
^       5.  Find  the  sum  of  7  terms  of  the  series  2,  —  |,  |,  •••. 

N  ^^'"-^^e.    Find  the  sum  of  12  terms  of  the  series  —  i?  i?  —  i> 

7.    Find  the  sum  of  7  terms  of  the  series  1,  2  x,  4  a^,  ••• 

r^  f  8.    Find  the  sum  of  7  terms  of  the  series  1,  —  2  ic,  4  a:^. 


r> 


.  •f     ,        9.    Find  the  sum  of  ii  terms  of  the  series  1,  ic^,  x*, 
't^^-'        10.    Find  the  sum  of  n  terms  of  the  series  1,  2,  4,  • 


356-  ACADEMIC  ALGEBRA 

11.  Find  the  sum  of  n  terms  of  the  series  1,  ^,  ^,  •••. 

12.  The  extremes  of  a  geometrical  series  are  1  and  729,  and  the 
ratio  is  3.     What  is  the  sum  of  the  series  ? 

13.  What  is  the  sum  of  the  series  3,  6,  12,  •.-,  192  ? 

14.  What  is  the  sum  of  the  series  7,  -  14,  28,  •-.,  -  224  ? 

371.   To  find  the  sum  of  an  infinite  geometrical  series. 

If  the  ratio  r  is  numerically  less  than  1,  it  is  evident"  that  the 
successive  terms  of  a  geometrical  series  become  numerically  less 
and  less.  Hence,  in  an  infinite  decreasing  geometrical  series,  the 
nth  term  I,  or  ar"~^,  can  be  made  less  than  any  assignable  number, 
though  not  absolutely  equal  to  zero. 

rl 


(III)  may  be  written   s  = 


1 


Since,  by  taking  enough  terms,  I  and  consequently  rl  can  be 
made  less  than  any  assignable  number,  the  second  fraction  may 
be  neglected. 

Hence,  the  formula  for  the  sum  of  an  infinite  decreasing  geomet- 
rical series  is 

a 


1-r 


(IV) 


Examples 

1.  Find  the  sum  of  the  series  1,  -^,  y^^,  •••. 

Solution 
Substituting  1  for  a  and  -^  for  r  in  (IV) , 

2.  Findthe  value  of  .185185185-... 

Solution 

Since  .185185185  .-.  =  .185  +  .000185  +  .000000185  + -..,  a  =  .185  and 
\   =.001. 

.185  5 


Substituting  in  (IV),     .185186185  ...  =  s 


1  -  .001      27 


Find  the  value  of 

3. 

i  +  i  +  i  +  -. 

-^4. 

3  +  I  +  A+" 

1]^- 

1-i+i--. 

PROGRESSIONS  357 


6.  .407407 

7.  .363636 


^>,    8.    1.94444 


372.   To  insert  geometrical  means  between  two  terms. 


Examples  ^6 

1.  Insert  3  geometrical  means  between  2  and  162.  c\\^ 

PROCESS  Explanation.  —  Since  there  are  three  means,  there  are  • 

7  _  fj,^n-i  five  terms,  and  n  —  1  =  4.    Solving  for  r  and  neglecting 

1  «9  _  9 ^  imaginary  values,  r=±S. 

IbZ  —  Zir         Therefore,  the  series  is  either  2,  6,  18,  54,  162,  or  2,  -6, 

r  =  ±  3  18,  -  54,  162. 

2.  Insert  3  geometrical  means  between  1  and  625. 

3.  Insert  5  geometrical  means  between  4J  and  -^^ff^. 

4.  Insert  4  geometrical  means  between  ^^  and  J^. 

5.  Insert  4  geometrical  means  between  5120  and  5. 

6.  Insert  4  geometrical  means  between  4V2  and  1. 

7.  Insert  5  geometrical  means  between  o^  and  h^. 

8.  Insert  6  geometrical  means  between  —  2  and  J  V2. 

9.  Insert  4  geometrical  means  between  a;  and  —  y. 

^  373.   Principle.  —  The  geometrical  mean  between  two  mimbers 
is  equal  to  the  square  root  of  their  product. 

The  above  principle  may  be  established  as  follows : 

Let  a  and  b  represent  any  two  numbers,  and  Gr  their  geometrical  mean. 
It  is  to  be  proved  that  G  =  Vab. 

Since  the  two  numbers  and  their  geometrical  mean  form  the  geometrical 
progression  a,  G^  6, 

i-m.  f=|, 

Q^  =  ab. 
.-.  G  =  y/ab. 


358-  ACADEMIC  ALGEBRA 

Find  the  geometrical  mean  between 

1.   8  and  50.  "4.    (a  +  bf  and  (a  -  bf. 

■^  2.   4  and  3|.  2,7.  j.  ,   7.2 

5     ^  +^^  and  ^^5±^. 
-  3.    Ill  and  f .  '    a^  _  ^5  ^5  _  ^2 

6.    25a^-10«+l  anda;2  +  i0aj  +  25. 

374.  Since  formulse  I  and  II,  or  III,  which  is  equivalent  to 
II,  are  two  independent  simultaneous  equations  containing  live 
elements,  if  three  elements  are  known,  the  other  two  may  be  found 
by  elimination. 

Problems 

375.  1.    Given  r,  I,  and  s,  to  find  a. 

2.  The  ratio  of  a  geometrical  progression  is  5,  the  last  term  is 
625,  and  the  sum  is  775.     What  is  the  lirst  term  ? 

3.  The  ratio  of  a  geometrical  progression  is  -^^,  the  sum  is  ^, 
and  the  series  is  infinite.     What  is  the  first  term  ? 

4.  Find  I  in  terms  of  a,  r,  and  s. 

5.  Find  the  last  term  of  the  series  5,  10,  20,  •••,  the  sum  of 
whose  terms  is  155. 

6.  If  i+  JV2  +  i  H =  IJ  (1  +  V2)?  what  is  the  last  term, 

and  the  number  of  terms  ? 

7.  Deduce  the  formula  for  r  in  terms  of  a,  I,  and  s. 

8.  If  the  sum  of  the  geometrical  progression  32  •••  243  is  665, 
what  is  the  ratio  ?     W^rite  the  series. 

9.  The  sum  of  a  geometrical  progression  is  700  greater  than 
the  first  term  and  525  greater  than  the  last  term.  V/hat  is  the 
ratio  ?     If  the  first  term  is  81,  what  is  the  progression  ? 

10.    Deduce  the  formula  for  r  in  terms  of  a,  n,  and  I. 

V  11.  The  first  term  of  a  geometrical  progression  is  3,  the  last 
term  is  729,  and  the  number  of  terms  is  6.  What  is  the  ratio  ? 
Write  the  series. 

12.    Find  I  in  terms  of  r,  n,  and  s. 


•t 


PROGRESSIONS  359 

^'  13.    The  sum  of  the  12  terms  of  a  geometrical  progression  whose 
ratio  is  2  is  4095.     What  is  the  12th  term  ? 

'  14.  The  velocity  of  a  sled  at  the  bottom  of  a  hill  is  100  feet 
per  second.  How  far  will  it  go  on  the  level,  if  its  velocity- 
decreases  each  second  \  of  that  of  the  previous  second  ? 

15.  From  a  cask  of  vinegar  \  was  drawn  off  and  the  cask  was 
filled  by  pouring  in  water.  Show  that  if  this  is  done  6  times, 
the  contents  of  the  cask  will  be  more  than  -f-^  water. 

16.  A  ball  thrown  vertically  into  the  air  100  feet  falls  and 
rebounds  40  feet  the  first  time,  16  feet  the  second  time,  and  so 
on.  What  is  the  whole  distance  through  which  the  ball  will 
have  passed  when  it  finally  comes  to  rest  ? 

'  17.  Show  that  the  amount  of  $  1  for  1,  2,  3,  4,  5  years  at  com- 
pound interest  varies  in  geometrical  progression. 

I  18.  Show  that  equimultiples  of  numbers  in  geometrical  pro- 
gression are  also  in  geometrical  progression. 

^^^9.    The  sum  of  three  numbers  in  geometrical  progression  is 
19,  and  the  sum  of  their  squares  is  133.     What  are  the  numbers  ? 

Suggestion.  —  When  there  are  but  three  terms  in  the  series  they  may  be 
represented  by  x^,  xy,  y^,  or  by  a;,  Va:?/,  y. 

20.  The  product  of  three  numbers  in  geometrical  progression 
is  8,  and  the  sum  of  their  squares  is  21.  What  are  the  three 
numbers  ? 

21.  If  4  is  a  geometrical  mean  between  two  numbers  whose 
sum  is  10,  what  are  the  numbers  ? 

22.  The  product  of  three  numbers  in  geometrical  progression 
is  64,  and  the  sum  of  their  cubes  is  584.    What  are  the  numbers  ? 

23.  The  sum  of  the  first  and  second  of  four  numbers  in  geo- 
metrical progression  is  15,  and  the  sum  of  the  third  and  fourth 
is  60.     What  are  the  numbers  ? 

Suggestion.  —  Four  unknown  numbers  in  geometrical  progression  may 

x^  w2 

be  represented  by  — ,  a;,  j/,  —  • 

24.  The  sum  of  the  first  and  third  of  three  numbers  in  geo- 
metrical progression  is  130,  and.  their  product  is  625.  What  are 
the  numbers  ? 


360  ACADEMIC  ALGEBRA 

25.  Divide  $700  among  three  persons  so  that  the  first  shall 
receive  $  300  more  than  the  third,  and  the  share  of  the  second 
shall  be  a  geometrical  mean  between  the  shares  of  the  first  and 
third. 

26.  If  a,  b,  and  c  are  in  geometrical  progression,  show  that 
their  reciprocals  also  are  in  geometrical  progression. 

27.  The  difference  between  two  numbers  is  24,  and  their 
arithmetical  mean  exceeds  their  geometrical  mean  by  6.  What 
are  the  numbers  ? 

HARMONICAL  PROGRESSION 

376.  1.  Examine  the  series  1,  ^,  ^,  ^,  •°°.  Has  it  a  constant 
difference  ?     Has  it  a  constant  ratio  ? 

2.  Take  the  reciprocal  of  each  term.  What  kind  of  a  series 
is  thus  formed?  How,  then,  may  the  series  1,  ^,  ^,  ^j  •••,  be 
described  ? 

377.  A  series  the  reciprocals  of  whose  terms  form  an  arith- 
metical progression  is  called  a  Harmonical  Series,  or  a  Harmonical 
Progression. 

3,  f ,  1,  f,  f,  ^,  •••  is  a  harmonical  progression,  because  ^,  -|,  1,  f,  f,  2,  ... 
the  reciprocals  of  its  terms  form  an  arithmetical  progression. 

H.  P.  is  an  abbreviation  for  the  words  Harmonical  Progression. 

378.  Problems  in  harmonical  progression  are  commonly  solved 
by  taking  the  reciprocals  of  the  terms  and  employing  the  prin- 
ciples of  arithmetical  progression.  There  is  no  general  method, 
however,  for  finding  the  sum  of  the  terms  of  a  harmonical  pro- 
gression. 

379.  Principle  1.  —  The  harmonical.  mean  between  two  numbers 
is  equal  to  twice  their  product  divided  by  their  sum. 

The  above  principle  may  be  established  as  follows : 
Let  H  represent  the  harmonical  mean  between  a  and  h. 
It  is  to  be  proved  that  H=^^^ 


a-\-h 


§377,  -,   — ,   -  are  in  arithmetical  progression. 

a    H    b 


PROGRESSIONS  361 


Hence,  §  356, 

1       1  _  1       1, 
h     H     H     a 

Clearing  of  tractions. 

aH-ah  =  ah-  hH. 

Transposing, 

aH-\-  bH='2ab. 

...  ^=2«^ 

a  +  6 

380.  Principle  2.  —  The  geometrical  mean  between  two  numbers 
is  also  the  geometrical  mean  between  their  arithmetical  and  harmoni- 
cal  means. 

The  above  principle  may  be  established  as  follows : 

§364,  ^  =  ^-  (1) 

§373,  G=:y/ab.  (2) 

§379,  j{=l^.  (3) 

a  -\-  b 

Multiplying  (1)  by  (3),  AH=  ab.  (4) 

Taking  the  square  root,  sJ~AH  =  yfab.  ,  (5) 

From  (2)  and  (6),  Ax.  1,  G^  =  VZff. 

Hence,  §  373,  G  is  the  geometrical  mean  between  A  and  H. 

Examples 

1.  Find  the  12th  term  of  the  H.  P.  6,  3,  2,  .... 

Solution.  — The  reciprocals  of  the  terms  form  the  arithmetical  progression 

\i  i»  \i  '" 
In  which  a  —  \  and  d  =  \. 

Substituting  \  for  a,  \  for  d,  and  12  for  n  in  (I), 

§369,  Z  =  i+(12-1H  =  2. 

Therefore,  §  377,  the  12th  term  of  the  given  harmonical  progression  is  \. 

2.  Find  the  10th  term  of  the  harmonical  series  \,  |,  i,  •••. 

3.  Insert  6  harmonical  means  between  \\  and  12. 

4.  Insert  2  harmonical  means  between  2  and  5. 

6.    Insert  7  harmonical  means  between  12^  and  2\. 


362  ACADEMIC  ALGEBRA 

6.  Insert  3  harmonical  means  between  b  and  a. 

7.  Find  the  nth  term  of  the  H.  P.  \,  |,  Jj-,  .... 

8.  The  3d  and  4th  terms  of  a  H.  P.  are  2^  and  1|-.     Write  the 
first  6  terms. 

Find  the  harmonical  mean  between 

/  9.   2  and  3.  13.  a  —  c  and  a-\-c. 

10.  ^  and  J.  14.  1  —  Va  and  1  +  Va. 

11.  2^  and  IJ.  15.  a  and  1 

12.  2iandl0.  13  V6andV3./ 

-^  17.   The  5th  term  of  a  harmonical  progression  is  ^,  and  the 
11th  term  is  ^^.     What  is  the  first  term  ? 

'   18.   The   arithmetical   mean   between  two  numbers  is  5,  and 
their  harmonical  mean  is  3^.     W^hat  are  the  numbers  ? 

19.   If  one  number  exceeds  another  by  2,  and  their  arithmetical 
mean  exceeds  their  harmonical  mean  by  ^,  what  are  the  numbers  ? 

■  20.    If  a,  b,  and  c  are  in  harmonical  progression,  prove  that 
a  —b  :h  —  c  =  a:  c. 

^21.    If  a,  b,  c,  and  d  are  in  harmonical  progression,  prove  that 
ab:  cd  =  b  —  a:d--c. 

22     If  b  is  the  harmonical  mean  between  a  and  c,  prove  that 


b  —  a      b  —  c     a     c 

23.  When  b  —  a:c  —  b  =  a\x,  prove  that  x  =  a,  if  a,  b,  and  c 
are  in  arithmetical  progression ;  that  x  =  b,  if  a,  6,  and  c  are  in 
geometrical  progression;  and  that  x  —  c,  if  a,  ^,  and  c  are  in 
harmonical  progression. 

24.  The  harmonical  mean  between  two  numbers  is  5^^  and  their 
arithmetical  mean  is  6^.     What  is  their  geometrical  mean  ? 

25.  Prove  that  x-^xy,  2xy,  and  xy -\- xy^  are  in  harmonical 
progression. 

,^  26.    If  6  H-  c,  c  +  a,  and  a-\-b  are  in  harmonical  progression, 
prove  that  a^,  h\  <?  are  in  arithmetical  progression. 


IMAGINARY   AND    COMPLEX   NUMBERS 


381.    1.    If  from  V—  25  the  rational  factor  V25  is  removed, 
what  irrational  factor  remains  ? 


2.  Simplify  V—  25,  V  —  16,  V  —  a^.  What  common  part,  or 
unit,  have  the  indicated  square  roots  of  negative  numbers  ? 

3.  What  is  the  square  of  V4?  of  V5?  of  V9?  of  V2^'? 
What  is  the  effect  of  squaring  a  radical  of  the  second  degree  ? 

What,  then,  is  the  square  of  V— 4?   of  V—  a?   of  V—  1  ? 

382.  Up  to  this  point  the  only  numbers  whose  nature  has  been 
discussed  have  been  numbers  that  differ  from  arithmetical  num- 
bers in  having  a  sign,  +  or  — ,  to  indicate  quality  or  direction. 
These  numbers  are  called  real  numbers,  and  may  be  briefly 
described  as  numbers  ivhose  squares  are  positive. 

There  are  numbers,  however,  ivhose  squares  are  negative.  They 
constitute  one  class  of  imaginary  numbers,  defined  in  §  257.  In 
this  chapter  only  imaginary  numbers  of  the  second  degree  are 
treated. 

Let  —  a  be  any  negative  real  number. 

Then,  V—  a  will  represent  an  imaginary  number. 

Since         +V—  a  =  + Va  •  V—  1  =  H- V—  1  •  Va 

and  —  V—  a  =  — Va  •  V—  1  =  — V—  1  •  Va, 

the  positive  imaginary  unit  -is  +  V—  1,  and  the  negative  imagi- 
nary unit  is  —V—  1. 

Since  the  square  of  the  square  root  of  a  number  is  the  number 
itself. 

This  relation  is  sufficient  to  explain  operations  with  imaginary 
numbers. 


T<-.B 


-1  ^+ 


A"\  ^ C 


•M 


864  ACADEMIC  ALGEBRA 

383.   Relation  between  the  units  +  1 ,  —  1 ,  V—  1,  and  —  V—  1. 

In  the  accompanying  figure  tlie  length  of  any  radius  of  the  circle  represents 
the  arithmetical  unit  1.     The  line  drawn  from  0  to  A,  called  the  line  OA, 
^  represents  the  positive  unit  +1,  and 

the  line  OA"  represents  the  nega- 
tive unit  —  1.  Every  real  number 
lies  somewhere  on  the  line  X'X, 
which  is  supposed  to  extend  indefi- 
nitely in  both  directions  from  0. 
' — -X  X'X  is  called  the  axis  of  real  num- 
bers. 

The  direction  of  any  line  drawn 
from  0,  as  OB,  that  is,  the  quality 
or  direction  sign  of  the   number 
represented  by  that  line,  is  deter- 
^  mined  with  reference  to  the  fixed 

line  OA  by  finding  what  part  of  a  revolution  is  required  to  swing  the  line 
from  the  position  OA  to  the  required  position.  By  common  consent  revo- 
lution of  the  line  OA  is  performed  in  a  direction  opposite  to  that  of  the  hands 
of  a  clock,  as  shown  by  the  arrows.  OB  is  reached  after  |  of  a  revolution, 
OA'  after  ^  of  a  revolution,  OA"  after  i  of  a  revolution,  etc. 

Since  OA"^  or  —1,  represents  |  of  a  revolution  of  OA,  the  square  of  OJ.", 
or  (—  1)2,  represents  1  revolution  of  OA,  which  produces  OA,  or  +  1. 
Hence,  OA",  or  —  1,  represents  the  square  root  of  +  1,  or  (+  1)^. 

Similarly,  since  OA'  represents  ^  of  |  of  a  revolution  of  OA,  and  OA'' 
represents  ^  of  a  revolution  of  OA,  OA'  represents  the  square  root  of  OA", 
or  of  -  1  ;  that  is,  OA'  =  V^^. 

If  OA"  i-s  swung  ^  of  a  revolution  from  the  position  OA"  to  the  position 
OA'",  OA"  will  be  multiplied  by  V—  1  just  as  OA  is  multiplied  by  V—  1  to 
produce  OA'.     Hence,  the  result  OA'"  =  -  1  •  V^H  =  -  V^^. 

+  1,  represented  by  OA,  and  —  1,  represented  by  OA",  are  the  units  for 
real  numbers,  that  is,  are  real  units.  Just  as  the  real  number  -f  a  is  repre- 
sented by  a  line  a  units  long  extending  from  0  toward  X,  and  the  real  num- 
ber —  a  by  a  line  a  units  long  extending  from  O  in  the  opposite  direction,  so 
the  imaginary  number  -|-  aV—  1,  or  (-}- V—  1)  x  a,  is  represented  by  a  line 
a  units  long  extending  from  0  toward  Y,  and  the  imaginary  number  — aV— 1, 
or  (— V—  1)  X  a,  by  a  line  a  units  long  extending  from  O  in  the  opposite 
direction,  toward  Y'.  Hence,  +V—  1  and  —  V—  1  are  the  units  for  imagi- 
nary numbers,  that  is,  they  are  imaginary  units;  -f  aV—  1  is  called  a  posi- 
tive imaginary  number  and  —  aV—  1  a  negative  imaginary  number. 

Y'  Y  is  called  the  axis  of  imaginary  numbers.  If  Y'  Y  is  taken  as  the 
axis  of  real  numbers,  then  X' X  becomes  the  axis  of  imaginary  numbers. 
Hence,  it  is  seen  that  imaginary  numbers  have  as  much  reality  as  real  num- 
bers.    Imaginary  numbers  were  named  before  their  nature  was  understood. 


IMAGINARY  AND   COMPLEX  NUMBERS 


365 


384.  In  the  graphical  illustration  of  the  relation  between  real 
and  imaginary  numbers  it  was  assumed  that  H-l-V— 1=V—  1, 
or  V—  1  •  1 ;  that  is,  the  Commutative  Law  for  multiplication  was 
assumed  to  apply  when  imaginary  numbers  were  involved.  It  is 
evident,  if  the  discussion  of  number  is  to  proceed,  that  in  any 
operation  imaginary  numbers  must  obey  all  the  laws  of  real  num- 
,  bers  except  those  which  determine  the  quality  of  the  result ;  and 
that  the  quality  of  the  result  is  determined,  as  far  as  the  imaginary 
numbers  are  concerned,  by  the  relation  (V—  1)^  =  —  1. 


1. 


385.  Powers  of 

(V^)^  =-1; 

(V^)*  =  (V^^)\v^^y  =  (_  1)  (_  1)  =  + 1 ; 

and  so  on.     Hence,  if  n  =  0  or  a  positive  integer, 

(V^i)*»+3=-v^^;  ( v^n:/"-^* = + 1. 

Hence,  a7iy  even  power  o/  V—  1  is  real  and  any  odd  power  is 
imaginary. 

386.  Operations  involving  imaginary  numbers. 


Examples 


Find  the  value  of 

1.    (V^«.        3.    (V^^". 

2.  (V^^y.    4.  (V^^)'^ 


5.  (V-iy«.    7.  (-■ 

6.    (V^^y.       8.    (-■ 


9.    Add  V—  a*  and  V— 16a*. 


Solution 


V-  a*  +  V-  16  a*  =■  a'V-  i  +  4  oV-  1  =  5  aV-  1. 


366  ACADEMIC  ALGEBRA 

Simplify  the  following : 
-10.    V-^-f-V-49.  13.    V^^^^n[2  +  4V^^. 


11.    V-9+V-64.  14.    5V-18-V^ 


12.    2V-4H-3V-1.  15.    3V-20-V-80. 

16.    (V^^  +  3V^  +  (V^a-3V^^). 


17.  {-\  —  9 ic^  —  V—  xy)  —  (V—  4 xy  +  V—  xy). 

18.  V^^  +  V^^4^  _  V-  a;^  +  3  xV^^. 


19.    V-l6-3V^^+V-18  4-V-o0+V-25. 


20.  V-8  +  aV-2-V-98-5V-2a=^. 

21.  V-  16  aV  +  V^^-'  -  V-  9  a^a^. 


22.    Vl  -  5  -  3vT^^10  +  2Vo  -  30. 


23.    Multiply  3V-10  by  2 V-  8. 

PROCESS 

3V^=30  X  2V^=^  =  3VlO V^I  X  2 VsV^I 


=  6V10x8x(-l) 

=  -  6  VSO  =  -  24  V5 

Explanation.  —  In  order  to  determine  the  sign  of  the  product,  each  imagi- 
nary number  is  reduced  to  the  form  bV  —  1.  The  numbers  are  then  multiplied 
together  as  oixlinary  radicals,  observing,  however,  that  V—  1  x  V—  1  =  —  1. 

24.  Multiply  V^^  +  3 V^^  by  4V^^-V^^. 

First  Solution     "  Second  Solution 

4V3^  -  x/"irT} =(4V2  -  \/8)V=n:  ■    w^^  -  v'^ 

(\/2  +  :3V:3)(4V2-V3)(-l)  _4V4-12V(5 

=  (8+ 12V6-\/6-9)(- 1)  +3\/9+       V6 

=  1-11V6  ~      _  iivo 

Multiply :  ■ 

25.  3V^=^  by  2V^=~15.  '  28.    SV^^T  by  V^^. 

26.  4V^^^^  by  V-12.  29.    V^^^  by  V^^IOS. 

27.  2V^^  by  5V^^.  30.    V=ni)()  by   V^^. 


IMAGINARY  AND   COMPLEX   NUMBERS 


367 


6  + 


31. 

32.    V-«&  +  V 


3  by  V^  6  -  V^=^. 


33.    V— ic.y  +  V- 


a  by  V— a6— V— ci. 
X  by  ^  —  xy  -\-^  —  x. 


-^ 


ex. 


34.  V-^-V-12  by  V-8-V-75. 

35.  V— a+V^^H-V^^  by  V^^  +  V— ^  —  V— c. 


4-  ^*1  VT^ 


36.    Divide    V— 12  by  V  — 3. 

Solution 

VT2 


\A=32 


1      Vl2 


V3 


=  V4=2. 


V-:3        V8>A^ 
37.    Divide    Vl2  by  V^^. 

Solution 
\/T2  >/l2  V4 


V-3      >/3^ 


2V-1 


=  -2\/^. 


38.    Divide  5  by  (V-  1)^ 


Solution 


(>/-l)8 

Divide : 

39.  V^n^  by  V^^. 

40.  V27  by  V^^. 

41.  UV^TS  by  2V'^. 

42.  —  V -  a-  by   V— 6^. 

43.  1  by   V^. 

44.  (V^)'-V^  by  V^. 

45.  V^  +  (V^)-  by  V^. 

46.  V8-3Vli  by  V^^. 

47.  V12+V3  by  V^. 

57.    V 


5(+l)   ^5(V-1)*^^^^3^^ 


(V-i)8     (V3-i)3 


48.  -2"  by  V^=n[.         -'  ^ 

49.  (V^=:i)^  by  JV^I. 

50.  (V^n[)'  by  (V^^)^^ 

51.  V4a6  by   V  —  6c. 


52.  V-20-V-2by  2V-1.    '^ 

53.  V^IlG- V^^  by  2V^^    ^ 

54.  (V^^)"  by   -  l-V^^. 

55.  (V^=n)^«  by  (V^=i:)-^. 


56.    V-a-+6V-l  by  V-a6 

4  by  V"=^- V^^- V'-=l.      ' 


ACADEMIC  ALGEBRA 

"^  387.    For  brevity  V  — 1  is  often  written  L 

Including  all  intermediate  fractional  and  incommensurable 
values,  the  scale  of  real  numbers  may  be  written 

..._3...-2...-1...0...  +  l...  +  2...  +  3...  (1) 

and  the  scale  of  imaginary  numbers,  composed  of  real  multiples 
of  +  i  and  —  i,  may  be  written 

3  4 2i ^..0...  +  i-"  +  2i...  +  3i...  (2) 

Since  the  square  of  every  real  number  except  0  is  positive  and 
the  square  of  every  imaginary  number  except  0  ^,  or  0,  is  negative, 
the  scales  (1)  and  (2)  have  no  number  in  common  except  0.    Hence, 

An  imaginary  number  cannot  he  equal  to  a  real  number  nor  cancel 
any  part  of  a  real  number. 

V  388.    The  algebraic  sum  of  a  real  number  and  an  imaginary 
number  is  called  a  Complex  Number. 

2  +  3  V  —  1,  or  2  +  Si,  and  a  -\-  b  V  — 1,  or  a  +  hi,  are  complex  numbers, 
a^  +  2  a&  V  —  1  —  62  is  a  complex  number,  since  a^  +  2  a6  V  —  1  —  &2  — 

(rt2  _  62)  +  2  ab  v^n:. 

^  389.    Two  complex  numbers  that  differ  only  in  the  signs  of 
their  imaginary  terms  are  called  Conjugate  Complex  Numbers. 

a  +  &  V  —  1  and  a  —  6  V  —  1,  or  a  -{-  bi  and  a  —  bi,  are  conjugate  com- 
plex numbers. 

*^    390.    The  su7n  and  product  of  two  conjugate  complex  numbers  are 
both  real. 

.  Let  a  +  &  V  —  1  and  a  —  6  V  —  1  be  conjugate  complex  numbers. 
Their  sum  is  2  a. 
Since  (V  -  1)2  =  —  l,  their  product  is, 

§  97,  a2  -  (6  V^n)2  =  a2  -  (  -  62) 

=  a2  +  1)2^ 

0    391.    If  two  complex  numbers  are  equal,  their  real  parts  are  equal 
and  also  their  imaginary  parts. 

Let  a  +  6  V-  1  =  x  +  y  V—  1. 

Then,  a  -  x  =  (y  -  h)  V^^, 

which,    §  387,  is  impossible  unless  a  =  x  and  y  =  b. 


IMAGINARY  AND    COMPLEX  NUMBERS 


369 


392.  i/*a  +  6V  —  1  =  0,  a  and  h  being  real,  then  a  =  0  and  6  =  0. 
For,  squaring,  a^  +  2  a&  V  —  1  —  6^  =  o, 

a2  _  52  ,3,  _  2  ah  V^H;, 
which,  §  387,  is  true  only  when  a  =  0  and  6  =  0. 

393.  Graphical  representation  of  a  complex  number. 

The  sum  of  3  positive  real  units  and  2  positive  imaginary  units  is  found  by 
counting  3  units  along  OX  in  the  positive  direction  from  0  and  from  that 
point,  i>,  measuring  2  units  upward  at 
right  angles  to  OX  in  the  direction  of 
the  axis  of  imaginary  numbers.  The 
line  OP  represents,  by  its  length  and 
direction^  the  combined  effect  or  sum 
of  the  directed  lines  OD  and  DP,  that 
is,  the  complex  number  3  +  2  i. 

The  same  result  may  be  obtained 
by  counting  2  units  along  OY  up- 
ward from  0  and  from  the  end  of 
the  second  division  measuring  3  units 
toward  the  right  at  right  angles  to 
OY  in  the  direction  of  the  axis  of  real  numbers.  Hence,  the  line  OP  repre- 
sents either  3  +  2  i  or  2  i  +  3. 

Similarly,  the  line  OP  represents  by  its  length  and  direction  2\  —  \i  or 
—  \i  +  2^,  and  the  line  OP"  represents  —  \  -\-  i  or  i  —  \. 

Represent  the  following  numbers  graphically : 

1.  3  +  4i.  3.    5  +  2?:.  5.   1  —  2. 

2.  2-3t.  4.   5-2i.  6.   4i-l. 

394.  Relation  of  complex  numbers  to  real  and  imaginary  numbers. 
Let  a  and  h  represent  any  real  numbers. 

In  the  figure  of  §  393  let  P  represent  any  point  a  units  dis- 
tant from  Y'  Y  and  b  units  distant  from  X'X. 

Then  OP,  or  the  complex  number  a -^  b  V  —  1,  represents  ariy 
number  whatever. 

If  P  lies  on  the  axis  of  real  numbers,  ft  =  0  and  the  complex 
number  a  +  &V— l  =  a,  a  real  number. 

If  P  lies  on  the  axis  of  imaginary  numbers,  a  =  0  and  the  com- 
plex number  a-\-b  V  —  1  =  ft  V  —  1,  an  imaginary  number. 

If  P  lies  in  both  axes,  a  =  0  and  ft  =  0,  and  the  complex  num- 
ber a^-b  V  — 1  =  0. 

ACAD.   AI.O.  — 24 


370 


ACADEMIC   ALGEBRA 


395.    Operations  involving  complex  numbers. 

Examples 
1.   Add  3  -  2 V^=i:  and  2 -h  5V^^. 

Solution 

3  _  2 v^31  +  2  +  SV^^n:  =  (3  +  2)  4- ( -  2  V^n  +  5 V^n^) 

Explanation. — Since,  §  387,  the  imaginary  terras  cannot  unite  with  the 
real  terms,  the  simplest  form  of  the  sum  is  obtained  by  uniting  the  real  and 
the  imaginary  terms  separately  and  indicating  the  algebraic  sum  of  the  results. 

Simplify  the  following : 

2.    (5+V^r4)H-(V^^-3). 
(2_V^;^T6)  +  (3+V^=^. 
(3- V^r8)  +  (4+V^T8). 
(V- 


3. 
4. 
5. 
6. 

7. 
8. 
9. 


IIO  -  Vl6)  +  (V-  45  +  V4). 
(4  4-  V^=~25)  -  (2  +  V^^). 
(3  -  2V^^)-(2  -  3V^^). 
(2  -  2V^=n:  +  3)-(Vl6  -  V^=T6). 


V-49-2- 

10.  Expand  (a  +  5V^^)(a  +  b 

Solution 
§91,  (a  +  6V^^)(a+ &V^n^)=a-  +  2a6V^^ +(6V 

§  384,  =  a2  4-  2  abV^^  -  h'^. 

11.  Expand  (V5  -  V^^)l 

Solution 
(V5_V33)2^  5_2V3ri5+(_3) 


1)^ 


Expand  the  following : 
12.    (2  +  3V^=T)(l4-V^. 

13.  (5-V^n:)(l-2V'^l). 

14.  (V2+V^(V8-V^. 


=  (5. 

-3)- 

-2V- 

-  15 

=  2- 

-2V- 

-15. 

15. 

(2 

+  3^2. 

16. 

(2 

-  3  i)\ 

11.  {a- uy 


IMAGINARY  AND   COMPLEX  NUMBERS 


871 


18.  Show  that  (14-V^=^)(1  +V^^^)(1 +V^=^)  =  -8.^^ 

19.  Show  that  (-  1  +  V^^)(-  1  +  V^^3)(-  1  +  V^^)  =8. 

20.  Show  that  (-|-f^V^(-KiV^)(-i-|-4V^)  =  l. 

21.  Divide  8  +  V^H^  by  3  +  2  V^H^. 


First  Solution 


8  + 


1=6+    V-l +2 

_  3  v^n:  +  2 

-  3V^^  +  2 


3+2>/-l 


v:rT 


The  real  terra  of  the  dividend  may  always  be  separated  into  two  parts,  one 
of  which  will  exactly  contain  the  real  term  of  the  divisor. 

Second  Solution 

8+\A=1^   (8+x/31)(8-2>/^)    ^26-13V^^^o     ^/ f 

3+2\/^T      (3  +  2V^l)(3-2\/^n[)  9  +  4 

Divide : 
^22.    3  by  1  -  V^=^.  26.    16  +-  4 V^=^  by  3  -  V^^. 

23.  2byl+V^^.  27.    a^  + // by  a  -  ftV^T. 

24.  4  4-V4  by  2-V^^.  28.    a -[- hi  hy  ai  +  h. 

25.  9  +- V^^  by  3  -  V^^.     29.    (1  +-  0^  by  1  -  ^.  ^ 
30.    Find  by  inspection  the  square  root  of  3  +-  2V—  10. 

Solution 

*3  +  2V^^l0=(5-2)+2V'5.  -2=  5  +  2  V5  •  -2 +(-2). 

.-.  V3+2V^^l0=V5  +  2V5Tir2+(_2)  =  >/5+  aA^. 
Find  by  inspection  the  square  root  of 

35.    12V~-^-5. 


31.    4+-2V-21.     33.    6 


-7. 


32.    1-I-2V-6.       34.    9H-2V-22.     36.    h^  +  2ah^-l-a\ 

37.  Verify  that  —  1  -f  V—  1  and  —  1  —V—  1  are  roots  of  the 
equation  a^+-2aj+-2=0. 

38.  Expand  (i  +  ^V^^)^ 


INEQUALITIES 


396.  One  number  is  said  to  be  greater  than  another  when  the 
remainder  obtained  by  subtracting  the  second  from  the  first  is 
positive,  and  to  be  less  than  another  when  the  remainder  obtained 
by  subtracting  the  second  from  the  first  is  negative. 

If  «  —  6  is  a  positive  number,  a  is  greater  than  h  ;  but  if  a  —  6  is  a  negative 
number,  a  is  less  than  h. 

Any  negative  number  is  regarded  as  less  than  0 ;  and,  of  two 
negative  numbers,  that  more  remote  from  0  is  the  less. 

Thus,  —  1  is  less  than  0  ;  —  2  is  less  than  —  1 ;  —  3  is  less  than  —  2  ;  etc. 

An  algebraic  expression  indicating  that  one  number  is  greater 
or  less  than  another  is  called  an  Inequality. 

397.  The  Sign  of  Inequality  is  >  or  <. 

It  is  placed  between  two  unequal  numbers  with  the  opening 
toward  the  greater. 

Thus,  a  is  greater  than  h  is  written  a>b;  a  is  less  than  b  is  written  a<h. 

The  expressions  on  the  left  and  right,  respectively,  of  the  sign 
of  inequality  are  termed  the  Jirst  and  second  members  of  the 
inequality. 

398.  When  the  first  members  of  two  inequalities  are  each 
greater  or  each  less  than  the  corresponding  second  members,  the 
inequalities  are  said  to  subsist  in  the  same  sense. 

When  the  first  member  is  greater  in  one  inequality  and  less  in 
another,  the  inequalities  are  said  to  subsist  in  a  contrai-y  sense. 

x'>a  and  y  >  6  subsist  in  the  same  sense,  also  x <  3  and  y  < 4  ;  but  x  >6 
and  y  <  a  subsist  in  a  contrary  sense. 

372 


INEQUALITTES  373 

399.  1.  If  2  is  added  to  each  member  of  the  inequality  8  >  5, 
how  will  the  two  inequalities  subsist  ?  How  will  they  subsist,  if 
2  is  subtracted  from  each  member  ? 

2.    Investigate  other  inequalities. 

Principle  1.  —  If  the  same  number  or  equal  numbers  are  added 
to  or  subtracted  from  both  nnembers  of  an  inequality,  the  resultiny 
inequality  will  subsist  in  the  same  sense. 

Let  a>b- 

Then,  §396,  a  —  b  =  ]j,  -a,  positive  number. 

Ad.liiig  c  —  c  =0,  Ax.  2,     a  -\-  c  —  (b  -\-  c)  =  p. 
Therefore,  a  +  c>b  -\-  c. 

400.  1.  What  is  the  effect  of  adding  2  to  each  member  of  the 
inequality  x  —  2>y?  What  is  the  effect  of  subtracting  2  from 
each  member  of  the  inequality  a  +  2  >  6  ? 

2.  If  a  term  is  transposed  from  one  member  of  an  inequality 
to  the  other,  what  must  be  done  to  its  sign  ? 

Principle  2.  —  A  term  may  be  transposed  from  one  member 
of  an  inequality  to  the  other,  provided  its  sign  is  changed. 
Let  a  —  6  >  c  —  d. 

Adding  6  to  each  side,  Prin.  1,  a>h  ■\-  c  -d. 

Adding  —  c  to  each  side,  Prin.  1,  «  —  c>6  —  d. 

401.  Principle  3.  —  If  the  signs  of  all  the  terms  of  an  ineq\iality 
are  changed,  the  resulting  inequality  will  subsist  in  a  contrary  sense. 

Let  a  -  b>c  —  d. 

Transposing  every  term,  Prin.  2, 

-c  +  d>  -  a  +  b] 

tliat  is,  -a-\-b<  —  c-\-d. 

402.  1.  If  both  members  of  the  inequality  10  >  8  are  multi- 
plied by  2,  how  will  the  two  inequalities  subsist  ?  How  will  they 
subsist,  if  both  members  are  divided  by  2  ? 

2.  How  will  they  subsist  in  each  case,  if  the  multiplier  or 
divisor  is  —  2  ? 


^74  JNEQUALfTIES 

Principle  4.  —  If  both  members  of  an  inequality  are  multi- 
plied or  divided  by  the  same  number,  the  resulting  inequality  will 
subsist  i7i  the  same  sense  if  the  multiplier  or  divisor  is  positive,  but 
in  the  contrary  sense  if  the  multiplier  or  divisor  is  negative. 

'     Let  a>b. 

Then,  §  396,  a  —  b  =p,  a,  positive  number. 

Multiplying  by  m,  ma  —  mb  =  mp. 

If  m  is  positive,  nip  is  positive, 

and  ma  >  mb. 

If  m  is  negative,  mp  is  negative, 

and  mb  >  ma,  or  ma  <  mb. 

Putting  —  for  m,  the  principle  holds  also  for  division. 
m 

403.  1.  If  the  corresponding  members  of  the  inequalities  Q>b 
and  4  >  2  are  added,  how  will  the  resulting  inequality  subsist  ? 
How,  if  —^>  —  Q  and  —  2  >  —  4  are  added  ? 

Principle  5.  —  If  the  corresponding  members  of  any  number  of 
inequalities  subsisting  in  the  same  sense  are  added  together,  the 
resulting  inequality  will  subsist  in  the  same  sense. 

Let  a^b,   C^d,   e>/,  etc. 

Then,  §  396,                        a  —  b,    c  —  d,    e  —  /,  etc. ,  are  positive  numbers. 
Hence,  their  sum,  a  +  c  +  e  +  •••  -(b  +  d  +  /+  •••)?  is  a  positive  number  ; 
that  is,  a  +  c  +  eH >b  +  d+f+  ■-. 

404.  Principle  6.  —  If  each  member  of  an  inequality  is  sub- 
tracted- from  the  corresponding  members  of  an  equation,  the  resulting 
inequality  icill  subsist  in  a  contrary  sense. 

Let  a  >  6  and  let  c  be  any  number. 

Then,  §  396,  «  —  6  is  a  positive  number. 

Since  a  number  is  diminished  by  subtracting  a  positive  number  from  it, 

c—{a  —  b)<,c. 

Transposing,  c  —  a<c  —  b. 

That  is,  if  each  member  of  the  inequality  a  >  6  is  subtracted  from  the  cor- 
responding member  of  the  equation  c  =  c,  the  result  is  an  inequality  subsisting 
in  a  contrary  sense. 


INEQUALITIES  '  375 

405.  It  is  evident  that  the  difference  of  two  inequalities  sub- 
sisting in  the  same  sense,  or  the  sum  of  two  inequalities  subsist- 
ing in  a  contrary  sense,  or  the  product,  or  the  quotient  of  two  in- 
equalities, member  by  member,  may  have  its  first  member  greater 
than,  equal  to,  or  less  than  its  second. 

For  example,  take  the  inequality  12  >  6, 

Subtracting  8  >  2,  or  adding  —  8  <  —  2,  the  result  is  4  =  4. 

Subtracting  8  >  1,  or  adding  —  8  <  —  1,  the  result  is  4  <  5. 

Multiplying  and  dividing  by  3  >  2,  the  results  are  36  >  12  and  4  >  3. 

Multiplying  by  —  2  >  —  4,  the  result  is  —  24  =  —  24.  Dividing  by  4  >  2, 
the  result  is  3  =  3. 

Multiplying  and  dividing  by  —  2  >  —  3,  the  results  are  —  24  <  —  18  and 
-6<-2. 

Examples 

406.  1.  Find  one  limit  of  x  in  the  inequality  3  x  —  10  >  11. 

Solution 
3x-l0>ll. 
Prm.  2,  Zx> 21. 

l*rin.  4,  «  >  7. 

Therefore,  the  inferior  limit  of  x  is  7  ;  that  is,  x  may  have  any  value  greater 
than  7. 

2.  Find  the  limits  of  x  in  the  simultaneous  inequalities 
3  a;  -1-  5  <  38  and  4  a;  <  7  a;  -  18. 

Solution 
3  X  -h  5  <  38.  '  (1) 

4  x  <  7  x  -  18.  (2) 

Transposing  in  (i) ,  Prin.  2,  3  x  <  33. 

.-.  Prin.  4,  x<ll. 

Transposing  in  (2),  Prin.  2,       -  3  x  <  -  18. 
Changing  signs,  Prin.  3,  3  x  >  18. 

.-.  Prin.  4,  x  >  6. 

The  inferior  limit  of  x  is  6,  and  the  superior  limit  is  11 ;  that  is,  the  given 
inequalities  are  satisfied  simultaneously  by  any  value  of  x  between  6  and  11. 


376  INEQUALITIES 

3.    Find  the  limits  of  x  and  ?/  in  3  a?  —  ?/  >  —  14  and  x-\-2y  =  0. 

Solution 
3x-y>-14.  (1) 

X  +  2  y  =  0.  (2) 

Multiplying  (1)  by  2,  Prin.  4, 

6  X  -  2  y  >  -  28.  (3) 

Adding  (2)  and  (3),  7  x  >  -  28. 

.-.  x>-4.  (4) 

Multiplying  (2)  by  3,  3  x  +  6  ?/  =  0.  (6> 

Subtracting  (5)  from  (1),  Prin.  1, 

-ly>-U. 
Dividing  by  -  7,  Prin.  4,  2/  <  2. 

That  is,  X  is  greater  than  —  4,  and  y  is  less  than  2. 

Find  the  limits  of  x  in  the  following : 

^    4.   6  a;  -  5  >  13.  |  4  x  -  11  >  i  a;, 

5.  5ic-l<6a;-f-4.  '    l  20  -  2  a;  >  10. 

6.  3  a;- la;  <  30.  r3-4a;<7, 


9. 

7.   4ajH-l<6a;-ll.  I  5  a;  +  10  <  20. 


10.    a;  +  ^4-^>25  and  <30. 
3         6 

11.   Find  the  limits  of  a;  in  a;^  _^  3  ^  -^  28. 

Solution  * 

x2  +  3x>28. 
Transposing,  Prin.  2,  x^  +  3  x  -  28  >  0. 

Factoring,  .         (x  -  4)  (x  +  7)  >  0. 

That  is,  (x  -  4)(x  +  7)  is  positive. 

If  (x  -  4)(x  +  7)  is  positive,  either  both  factors  are  positive  or  both  are 
negative.  Both  factors  are  positive,  when  x  >  4  ;  both  factors  are  negative, 
when  X  <  —  7. 

Therefore,  x  can  have  all  values  except  4  and  —7  and  intermediate  values. 


INEQUALITIES  377 

Find  the  limits  of  x  in  the  following : 

12.  a^4-3a;>10.  16.    a^  >  9  ic  -  18. 

13.  a^  +  8aj>20.  17.    a^ -f- 40cc>  3(4aj  -  25). 

14.  ar^  +  5a;>24.  18.    m? -\- hx  >  ax -{- ah. 

15.  (a;-2)(3-a;)>0.  19.    (a;  -  3)  (5  -  a;)  >  0. 

Find  the  limits  of  x  and  y  in  the  following,  and,  if  possible, 
one  positive  integral  value  for  each  unknown  number : 


2a;-32/<2, 
20.    1  ^         '  23. 

^  2a; +  52/ =  25. 


21. 


22. 


2/  =  3a;-t-4, 
,25<22/  +  3a?. 

f  3a; +  2?/ =  42,  f2,_a;>9, 


x  +  y  =  10y  ^^     r  a;  >  2/  -h  4, 

4a;<3  2/.  *la;-22/  =  8. 


If  a,  ?;,  and  c  are  positive  and  unequal, 

26.  Which  is  the  greater,    ^  +  ^    or  ^L±A^? 

^  '  a  +  26        a  +  36 

27.  Prove  that  a^  +  6-  >  2  a6. 

Suggestion.  —  Whether  a  —  6  is  positive  or  negative,  (a  —  6)^  is  positive. 

28.  Prove  that  a^  -\- h"^  +  (?  >  ab -\- ac  +  he. 

29.  Prove  that  a^-{-h^>  a^h  +  oIP. 

30.  Which  is  the  greater,  ^?L±_^  or  ^-±-^? 

31.  Prove  that  —  -f  —  >  1,  except  when  2  a  =  3  6. 

36     4a 

32.  Prove  that  (a  —  2  5)  (4  6  —  a)  <  b'\  except  when  a  =  'dh. 

33.  Prove  that  a^ +  W  +  (?>'dabc. 

34.  Prove  that  the  sum  of  any  positive  real  number,  except  1, 
and  its  reciprocal  is  always  greater  than  2. 


VARIABLES   AND    LIMITS 


>     407.    A  number  that  has  the  same  value  throughout  a  discussion 
is  called  a  Constant. 

Arithmetical  numbers  are  constants.  A  literal  number,  as  a  or  ic,  is  con- 
stant in  a  discussion,  if  it  keeps  the  same  value  throughout  that  discussion. 

X    408.    A  number  that  under  the  conditions  imposed  upon  it  may 
have  a  series  of  different  values  is  called  a  Variable. 
Variables  are  usually  represented  by  x,  y,  z,  etc. 

The  numbers  .3,  .33,  .333,  .3333,  ...  are  successive  values  of  a  variable 
approaching  the  constant  -|. 

The  commensurable  numbers  1,  1.4,  1.41,  1.414,  1.4142,  ...  are  successive 
values  of  a  variable  approaching  the  constant  y/2. 

^'  409.    An  expression  whose  value  depends  upon  the  value  of  a 
variable  is  called  a  Function  of  that  variable. 

2x4-1  is  a  function  of  x,  because,  if  successive  values  are  given  to  x, 
2  X  -f  1  will  take  successive  values.  For  example,  if  x  =  0,  2  x  +  1  =  1  ;  if 
x=l,  2x  +  l=3;  ifx  =  2,  2x+l  =  5,  etc.  x^  -  2  x  and  —1—  are 
also  functions  of  x.  1  —  x 

x'^  +  2  x^  —  5  ^^  is  a  function  of  x  and  z. 

Every  function  of  a  single  variable  is  a  variable. 

The  variable  to  which  different  values  may  be  given  at  pleasure 
or  according  to  some  law  is  called  the  Indepeyideyit  Variable,  and 
the  function  of  the  independent  variable  is  called  the  Dependent 
Variable. 

In  the  first  illustration  x  is  the  independent  variable  and  2x+l,  the 
function  of  x,  is  the  dependent  variable. 

410.  To  discuss  functions  of  a  variable  it  is  necessary  to  sup- 
pose that  the  variable  takes  its  successive  values  according  to  some 
definite  law  of  change. 

378 


VARIABLES  AND  LIMITS  379 

Where  a  variable  takes  a  series  of  values  that  approach  nearer 
ana  nearer  a  given  constant,  so  that  by  taking  a  sutticient  number 
of  steps  the  difference  between  the  variable  and  the  constant  can 
be  made  numerically  less  than  any  conceivable  number  however 
small,  the  constant  is  called  the  Limit  of  the  variable,  and  the 
variable  is  said  to  approach  its  limit. 

The  variable  .3,  .33,  .333,  .3333,  .  .  .,  whose  increase  at  each  step  is  ^^  of 
the  previous  increase,  approaches  ^  as  a  limit.  For  .3  differs  from  i  by  less 
than  ^j,  .33  by  less  than  y^^,  .333  by  less  than  x^Vir»  ^^^-i  ^^^  ^y  taking  a 
sufficient  number  of  steps  it  is  possible  to  obtain  a  value  of  the  variable  differing 
from  I  by  less  than  any  number  that  can  be  conceived  of,  however  small. 

411.  The  difference  between  a  variable  and  its  limit  is  a  variable 
whose  limit  is  zero. 

As  .3,  .33,  .333,  .  .  .  approaches  its  limit  i,  the  variable  difference  |  —  .3, 
^  —  .33,  ^  —  .333,  .  .  .f  OT  ■^,  j^,  Tihui  '  ■  '  approaches  the  limit  zero. 

A  variable  may  approach  a  constant  without  approaching  it  as 
a  limit. 

The  variable  6.6,  6.66,  6.666,  .  .  . ,  in  approaching  6f  as  a  limit,  approaches 
7  also,  but  not  as  a  limit. 

A  variable  in  approaching  its  limit  may  be  always  less  than  its 
limit,  or  always  greater,  or  sometimes  greater  and  sometimes  less. 

The  variable  .3,  .33,  .333,  ...  is  always  less  than  its  limit  ^. 

The  variable  .7,  .67,  .667,  ...  is  always  greater  than  its  limit  1  —  |. 

The  sum  of  the  first  n  terras  of  the  geometrical  series  1,  —  i,  J,  —  i,  t^, 
—  y\,  ...  is  a  variable  whose  successive  values  1,  ^,  |,  f,  |^,  f^,  .  .  .  are 
alternately  greater  and  less  than  the  limit  f. 

A  variable  may  increase  or  decrease  according  to  its  law  of 
change  and  become  numerically  greater  or  less  than  any  assignable 
number. 

The  variable  1,  —  2,  4,  —  8,  16,  .  .  .  may  become  numerically  greater  than 
any  number  that  can  be  assigned.  The  variable  1,  ^,  ^,  |,  i^g,  •  .  •  may 
become  numerically  less  than  any  number  that  can  be  assigned. 

"'''412.    A  variable  that  may  become  numerically  greater  than  any 
assignable  number  is  said  to  be  Infinite. 
The  symbol  of  an  infinite  number  is  go. 

^  413.    A  variable  that  may  become  numerically  less  than  any 
assignable  number  is  said  to  be  Infinitesimal. 
An  infinitesimal  is  a  variable  whose  limit  is  zero. 


380  VARIABLES   AND  LIMTTS 

The  character  0  is  used  as  a  symbol  for  an  infinitesimal  number 
as  well  as  for  absolute  zero,  the  result  obtained  by  subtracting  a 
number  from  itself. 

414.  A  number  that  cannot  become  either  infinite  or  infinitesi- 
mal is  said  to  be  Finite. 

415.  Interpretation  of  -• 

If  the  numerator  of  the  fraction  -  is  constant  while  the  de- 
it' 

nominator  decreases  regularly  until  it  becomes  numerically  less 
than  any  assignable  number,  the  quotient  will  increase  regularly 
and  become  numerically  greater  than  any  assignable  number. 

.-.  ^  =  00.     That  is. 

If  a  finite  number  is  divided  by  an  infinitesimal  numbeVy  the 
quotient  will  be  an  infinite  number. 

•  416.    Interpretation  of  — 

oo 

If  the  numerator  of  the  fraction  -  is  constant  while  the  de- 

X 

nominator  increases  regularly  until  it  becomes  numerically  greater 
than  any  assignable  number,  the  quotient  will  decrease  regularly 
and  become  numerically  less  than  any  assignable  number. 

.-.  -  =  0.     That  is, 

00 

If  a  finite  number  is  divided  by  an  infinite  number,  the  quotient 
will  be  an  infinitesimal  number. 

417.  The  symbol  =  is  read  ^  approaches  as  a  limit  J 
The  abbreviation  lim.  is  used  for  limit. 

x  =  a  is  read  '  x  approaches  a  as  its  limit. '  The  expression  lim.  x  =  a  is 
equivalent  to  x  =  a,  and  is  read  '  the  limit  of  a-  is  a.' 

Though  X  represents  a  variable  that  may  transcend  all  finite 
values,  it  is  convenient  to  use  the  expression  a;  =  oo  to  indicate 
that  X  increases  numerically  without  limit. 

Thus,  as  X  =  00,  -  =  0. 

X 

418.  Prtxciple  1. — A  variable  cannot  approach  two  unequal 
limits  at  the  same  time. 


VARIABLES  AND  LIMITS  381 

For  in  approaching  the  more  remote  as  a  limit  the  variable  would  reach  a 
value  intermediate  between  the  two,  and  thereafter  in  approaching  one  as 
a  limit  it  would  recede  from  the  other,  which,  therefore,  could  not  be  a  limit. 

419.  Principle  2.  —  If  two  variables  are  always  equal  and  each 
approaches  a  limit,  their  limits  are  equal. 

For  in  their  values  the  two  variables  are  but  one. 

Hence,  Prin.  1,  the  limit  of  their  common  values  is  their  common  limit. 

420.  1.  Since  the  limit  of  .333  •••  is  |,  what  is  the  limit  of 
2 +  .333  ...,  or  2.333-..?  of  4 +  .333,  or  4.333...?  of  5.333...? 
of  2-.333...,  or  1.666...?   of  .333 .13,  or  .203...? 

2.  What  is  the  limit  of  the  algebraic  sum  of  a  constant  and  a 
variable  ? 

Principle  3.  —  TJie  limit  of  the  algebraic  sum  of  a  constant  and 
a  variable  is  the  algebraic  sum  of  the  constant  and  the  limit  of  the 
variable. 

'  421.   1.   Since  the  limit  of  .333-..  is  ^,  what  is  the  limit  of 
M^'"?  of  .111  ...? 

2.  Since  the  limit  of  1  +^  +  }+  ...  is  2,  what  is  the  limit  of 
3  +  f  +  i+.-?   of  T-V  +  ^V  +  A  +  -? 

3.  How  may  the  limit  of  the  product  of  a  variable  and  a  con- 
stant be  obtained  from  the  limit  of  the  variable  ? 

Principle  4.  —  The  limit  of  the  product  of  a  variable  and  a 
finite  constant  is  equal  to  the  product  of  the  constant  and  the  limit 
of  the  variable. 

The  above  principle  may  be  established  as  follows: 

Case  1. —  When  the  limit  of  x  is  0. 

Let  k  be  any  finite  constant. 

It  is  to  be  proved  that  kx  =  0. 

However  small  any  number,  as  q,  may  be,  since  x  =  0,  x  may  be  made 
numerically  less  than  q-r-k. 

Hence,  kx  may  be  made  numerically  less  than  q  ;  that  is,  kx  may  be  made 
numerically  less  than  any  number  however  small. 

Therefore,  §  410,  kx  =  0. 

Case  2.  —  When  the  limit  of  x  is  not  0. 

Let  k  be  any  finite  constant,  a  the  limit  of  x,  and  y  the  variable  that  must 
be  added  to  x  to  produce  Q. 


382  VARIABLES  AND  LIMITS 

It  is  to  be  proved  that  kx  =  ka. 

Since  ic  +  y  =  a,  x  —  a  —  y\ 

.*.  kx  =  ka  —  ky. 
Prin.  2,  lim.  {kx)  =  liiu.  (ka  -  ky) 

Prin.  3,  =  ka  -  lim.  {ky). 

But  since,  §  411,  y  =  0,  by  Case  1,  lim.  ky  =  0. 

Hence,  lim.  (kx)  =  ka  —  0  =  ka  ; 

that  is,  kx  =  ka. 

^NoTE.  — The  principle  holds  for  the  limit  of  a  variable  divided  by  a  con- 
stant, since  dividing  by  k  is  equivalent  to  multiplying  by  — 

k 

422.  Principle  5.  —  The  limit  of  the  variable  sum  of  any  finite 
number  of  variables  is  equal  to  the  sum  of  their  limits. 

The  above  principle  may  be  established,  as  follows : 

Let  x:^  a,  y  =  b,  z  =  c,  etc. ,  to  any  finite  number  of  variables,  as  n. 

It  is  to  be  proved  that  lim.  (x  -\-  y  -i-  z  -\-  •••)  =  a  +  b  -\-  c  +  '•-. 

Let  vi,  U2,  Vs,  '•-  be  the  variable  differences  between  x,  y,  z  ■■•  and  their 
respective  limits. 

Then,  x  +  y  +  z  -^  -  =  (a  -{-  b  +  c  +  •■')-(vi -\- V2 -h  Vs  +  ..-), 

and       lim.  (x  +  y  +  z  +  •••)  =  lim.  [(a-\-b-t  c  +  ••.)  -  (vi  +  vo  +  vs  +  •••)] 

Prin.  3,  =  a  -\-  b  +  c -\ lim.  (vi  +  V2  +  Us  +  •••)• 

Since,  §411,  ■«!  =  0,  V2  —  O,  vs  =  0,  etc.,  however  small  any  number,  as 
q,  may  be,  each  of  the  n  variables,  vi,  v^,  vs,  etc.,  may  be  made  less  than 
q  -^n,  and  hence  their  sum  may  be  made  less  than  q. 

Therefore,  §  410,  lim.  (vi-\- V2  +  vs -\- '-)  =  0. 

Hence,  lim.  (x-\-y-\-  z-\ — )=  a  +  b-\-c-\ — . 

■"  423.   Principle  6.  —  Tlie  limit  of  the  variable  product  of  two  or 
more  variables  is  equal  to  the  product  of  their  limits. 

The  above  principle  may  be  established  as  follows: 
Given,  x  =  a  and  ?/  =  b. 
It  is  to  be  proved  that    lim.  (xy)  =  ah. 
Let  v\  =  a—x  and  V2  =  b  —  y. 

Then,  xy=  (a -vi)(b -V2), 

and  .      lim.  (xy)  —  lim.  [ab  -  bvi  —  av^  4-  Viv^'] 

Prin.  5,  Prin.  4,  =ab  —  b  lim.  vi  —  a  lim.  v^  +  lim.  (V1V2) 

§411,  =;a6+lim.  (vi7;2). 


VARIABLES  AND   LIMITS  383 

Since  when  vi  and  V2  are  near  their  common  limit  0,  their  product  is  much 
less  tlian  either  vi  or  vg, 

lim,(i7iV2)=0. 

Hence,  lim.  (xy)  =  ab  +  0  =  ab. 

Similarly,  the  principle  may  be  established  for  any  number  of  variables. 

424.  Principle  7.  —  The  limit  of  the  variable  quotient  of  two 
variables  is  equal  to  the  quotient  of  their  limits,  provided  the  limit 
of  the  divisor  is  not  0. 

The  above  principle  may  be  established  as  follows : 

y 
Let  X  —  '-  :  and  let  lim.  z  be  not  0. 

z  ' 

It  is  to  be  proved  that  lim.  x  —  ""'  ^- 


_,  .~  —  ^„  ^ . —  — 

lim.  z 

Since  ic  =  ^» 

y  =  xz. 

Prin.  6, 

lim.  y  =  lim.  x 

Therefore, 

Um.x=^}'"y. 

lim.  z 

The  principle  has  no  meaning  when  z  =  0,  since  lim.  y  cannot  be  divided 
byO. 

425.  When  by  causing  a  variable  x  to  approach  sufficiently 
near  to  a  it  is  possible  to  make  the  value  of  a  given  function  of  x 
approach  as  near  as  we  please  to  a  finite  constant  I,  I  is  called  the 
ilmit  of  the  function  when  x  =  a. 

Suppose  that  .1,  .11,  .111,  .1111,  •••  are  successive  values  of  x  approaching 
J  as  a  limit.  Then,  the  corresponding  values  of  1  —  2x,  a  function  of  x,  are 
.8,  .78,  .778,  .7778,  values  of  a  variable  approaching  ^  as  a  limit.  As  a;  =  ^, 
the  function  of  x,  1  —  2  x,  =  | ;  for  by  causing  x  to  approacli  sufficiently 
near  to  ^  it  is  possible  to  make  1  —  2  a:  approach  as  near  as  we  please  to  J. 

The  expression 

lim.  [function  of  x"]^^^ 

is  read  ^  limit  of  function  of  a:  as  a;  approaches  a  as  a  limit.' 

Thus,  lim.  (1  —  2x),.^^  =  |  indicates  that  as  x  approaches  its  limit  ^, 
1  —  2  x  approaches  the  limit  I. 

7^426.  In  finding  the  limiting  values  of  the  functions  given  in 
the  following  examples,  the  student  is  expected  to  apply  the 
principles  that  have  been  established  above. 

Finding  the  limiting  value  of  a  function  of  aj  as  ic  =  a  is  called 
evaluating  the  function  for  x  =  a. 


384  VARIABLES  AND   LTMITS 

Examples 

If  x  =  a,  y  =  2,  and  2;  =  0,  find  the  limit  of 

1.  x-{-y  -{-  z.  ^.   x  —  \y-\-ax  —  y^. 

2.  axy  —  a?.  5.    (x  -\-  y)  x  —  {x  —  y) z. 

3.  ^-U.  6.   ^L±^±^ _|_ £_±1. 
'2a  '       x—y  a 

Find  the  value  of 


7.  Lim.r^-^"  +  n 

Lcc^  +  r^  +  l  J;t=-1 


.  427.   Indeterminate  forms  -,  — ,  0  x  co,  oo  x  0,  00—00. 

^  0     QO 

For  all  values  of  a  and  a?, 


X     a     X 


\~  X 


j  =  a.  (1) 


If  ic  =  Qc,  (1)  becomes  -  =  0  x  00  =  a. 
If  a:  =  0,  (1)  becomes  —  =  00  x  0  =  a. 

GO 

Since   a   denotes   any   number   whatever,   -,  — ,   0  x  00,    and 

0      QO 

00  X  0  are  symbols  for  indeterminate  numbers. 

If  k  is  any  constant,  ooj  -|-  Zc  =  002 ;    .-.  X2  —  oOj  =  fc. 
Henca,  00  —  oo  is  a  symbol  for  an  indeterminate  number. 


VARIABLES  AND  LIMITS  385 

^428.  Since  every  function  of  a  single  variable  is  a  variable,  it 
is  evident  that  the  preceding  principles  apply  to  functions  of  a 
variable.  Thus,  to  apply  to  functions  of  a  variable,  Prin.  5,  6,  and 
7  may  be  stated  as  follows : 

The  limit  of  the  sum  of  a  finite  number  of  functions  of  x  is  equal 
to  the  sum  of  their  limits. 

The  limit  of  the  product  of  a  finite  number  of  functions  of  x  is 
equal  to  the  product  of  their  limits. 

The  limit  of  the  quotient  of  two  functions  of  x  is  equal  to  the 
quotient  of  their  limits,  provided  the  limit  of  the  divisor  is  not  zero. 

These  principles  fail  to  give  a  limit  whenever  the  result 
obtained  involves  one  of  the  indeterminate  forms, 

00  —  00,   0  X  00,   00  X  0,    -,    — 
0     00 

^  429.  The  preceding  principles  of  limits  lead  to  the  conclusion 
that  the  limit  of  a  function  is  found  by  substituting  the  limits  of 
the  variables  for  the  variables,  except  when  such  a  substitution 
gives  an  indeterminate  form  (§  428). 

Thus,  if  lim.  x  =  5  and  lim.  y  =  2,  the  limit  of  4  x  —  3  y  is  found  by  substi- 
tuting 5  for  X  and  2  for  y  in  the  function  4  a;  —  3  y.    But  if  substitution  is 

employed  directly  to  evaluate  the  functions ,  -  x  y,  and 

x-\     «  (x  -  1)    y 

(x2  —  l)^(x  -  1)  when  x  =  1  and  y  =  0,  these  functions  take  the  forms 

oo  —  Qo,  Qo  X  0,  and  0  h-  0,  respectively. 

When  the  method  of  evaluation  by  substitution  in  the  given 
function  fails,  the  evaluation  of  the  function  is  performed  by  the 
aid  of  Prin.  2. 

Thus,  to  evaluate  (x^  -  1)  ^  (x  -  1)  when  x  =  1,  find  another  function  of 
X,  as  X  +  1,  equal  to  the  given  function  (x^  —  1) -^  (x  —  1)  for  all  values 
assumed  by  x  while  approaching  the  limit  1. 

If  X  takes  the  successive  values 

2,  !,  f,  f,  Hi  !i  -M  approaching  1, 

then  both  functions  (x^  —  1)  h-  (x  —  1)  and  x  +  1,  take  the  successive  values 

3,  I,  I,  V.  fl.  Ml  •••»  approaching  2. 

Since  the  two  functions  are  equal  for  all  values  of  x  as  x  approaches  its 
limit  1,  by  Prin.  2  they  have  the  same  limit.  This  limit  is  lim.  (x  +  l)xii, 
which  by  substituting  lim.  x  for  x  is  found  to  be  1  +  1,  or  2. 

ACAD.  ALG.. —  25 


386  VARIABLES  AND  LIMITS 

Examples 
Find  the  value  of 

lof  -  4jx=2  L^  +  a'jx=-a 

2.   Li..p-^^  4.   Lim.r4±i:i 

|_1  -  x_}x=l  [_xr  -  6jx=-6 

^     J.      [x'-Bxi-ei 

L  0^-8x^  +  7  J,=^i 

7.   Find  the  limiting  value  of  - — -^ —^ — — —  when  a;  =  0 

and  also  when  x  =  cc.  '  .     ' 


Solution. — As  x  approaches  the  limit  0,  the  first  three  terms  of  the 
numerator  and  also  of  the  denominator  become  infinitely  small  as  compared 
with  the  fourth,  and,  consequently,  may  be  neglected.  Hence,  when  x  =  0, 
the  fraction  approaches  the  limiting  value  \. 

As  aj  =  Qo,  that  is,  as  x  becomes  indefinitely  greater,  the  last  three  terms 
of  the  numerator  and  also  of  the  denominator  become  infinitely  small  as 
compared  with  the  first,  and,  consequently,  may  be  neglected.  Hence,  when 
x  =  CO,  the  fraction  approaches  the  limiting  value  — ^,  or  — 

Find  the  limiting  values  of  the  following  when  aj  =  0  and  when 
l+a^  +  ^'  +  a^  ,o        5a^  +  10       ^^"^ 


8.    z — '         '         '        •  12. 


13. 


10.    X-; r :  -  14. 


11.    ^ -^^ --—^.  15. 


1-a^ 

''-x'-2x^ 

5a^- 

x'  +  4.x  +  2 

2a^-j-3x^-x-{-l 

4.x' 

-3ar^  4-^  +  1 

2x'- 

a^-a^  +  «?  +  l 

2x'- 

3a^-h2r^-2 

x^  +  2x  +  2 

3a;-4 

x'-x-^' 

2aj-l 

4a^  +  5a^  +  2a;  ^ 


x^-2x'  +  x-{-l  2a^  +  a;  +  l 


INTERPEETATION   OF   EESULTS 


430.  When  the  roots  obtained  by  solving  an  equation  satisfy 
the  equation,  the  solution  has  been  properly  performed ;  but  the 
results  found  in  solving  a  problem  may  sometimes  be  at  variance 
with  some  condition  of  the  problem.  Consequently,  the  inter- 
pretation of  results  becomes  important. 

POSITIVE    RESULTS 

431.  Since  the  numbers  sought  in  the  solution  of  a  problem 
are  arithmetical  rather  than  algebraic,  when  positive  results  are 
obtained,  it  is  not  likely  that  they  will  conflict  with  the  condi- 
tions of  the  problem.  Sometimes,  however,  even  a  positive  result 
violates  one  or  more  of  the  conditions  of  a  problem. 

In  such  cases  the  problem  is  impossible. 

432.  1.  A  club  consisting  of  25  members  raised  the  sum  of 
$  13  by  assessing  the  men  80  cents  each  and  the  women  40  cents 
each.     How  many  men  were  there,  and  how  many  women  ? 


Solution 

Let 

X  =  the  number  of  men. 

Then, 

25  -  X  =  the  number  of  women 

••• 

fx  +  K25-x)=13; 

whence, 

x  =  7h 

and 

25  -  X  =  17i 

Though  the  numbers  found  will  satisfy  the  equation,  yet  since 
the  number  of  men  and  the  number  of  women  cannot  be  frac- 
tional, the  problem  is  impossible. 

387 


388  INTERPRETATION  OF  RESULTS 

2.  The  second  digit  of  a  number  expressed  by  two  digits  is 
twice  the  first,  and  4  times  the  first  digit  is  9  greater  than  the 
second.     What  is  the  number  ? 


Solution 

Let 

X  =  the  first  digit. 

Then, 

2  a:  =  the  second  digit ; 

.*. 

4ic  =  2x  +  9; 

whence, 

X  =  4^,  the  first  digit, 

and 

2x  =  9,  the  second  digit. 

While  these  numbers  satisfy  the  equation,  they  fail  to  satisfy 
the  implied  condition  that  the  digits  must  be  integers. 
Hence,  the  problem  is  impossible. 

NEGATIVE    RESULTS 

433.  A  few  examples  will  suggest  the  methods  to  be  employed 
in  the  interpretation  of  negative  results. 

1.  If  A  is  40  years  old  and  B  is  30,  in  how  many  years  will  A 
be  twice  as  old  as  B  ? 

Solution 

Let  X  =  the  number  of  years  before  A  will  be  twice 

as  old  as  B. 

Then,  40  +  a;  =  2  (30  +  x)  ; 

whence,  a;  =  —  20. 

Though  the  result  is  algebraically  correct,  inasmuch  as  —  20 
substituted  for  x  satisfies  the  equation,  nevertheless  it  is  arith- 
metically absurd.  Hence,  the  conditions  of  the  problem  are 
inconsistent  with  each  other.  Had  the  result  been  +  20,  the 
statement  that  A  would  be  twice  as  old  as  B  in  20  years  would 
have  been  arithmetically  reasonable.  However,  since  —  x=  -\-  20, 
the  equation  will  give  a  result  arithmetically  reasonable,  if  —  a?  is 
substituted  for  x;  that  is,  if  x  is  taken  to  represent  the  number  of 
years  since  A  was  twice  as  old  as  B. 

The  conditions  of  the  problem  should,  therefore,  be  modified  as 
follows :  If  A  is  40  years  old  and  B  is  30,  how  many  years  agQ 
w^s  A  twice  as  old  as  B  ? 


INTERPRETATION  OF  RESULTS  389 

2.    How  much  money  has  A,  if  i  of  his  money  is  5  dollars  more 


an  i  of  it? 

Solution 

Let 

X  =  the  number  of  dollars  A  has. 

Then 

4      3 

Solving, 

x=-60. 

While  the  result  —  60  satisfies  the  equation,  it  violates  the  sup- 
position, made  in  the  problem,  that  A  has  some  money. 
If  —  a;  is  substituted  for  x,  the  equation  becomes 


X 


T=  5 ;  whence,  x  =  60. 


The  problem  when  modified  to  express  conditions  arithmetically 
reasonable  will  be  :  How  much  money  has  A,  if  i  of  it  is  5  dollars 
less  than  ^  of  it ;  or,  if  —  60  dollars  is  interpreted  60  dollars  in 
debt:  How  much  money  does  A  owe,  if  J  of  what  he  owes  is  5 
dollars  more  than  ^  of  it  ? 

434.  From  the  above  discussions  we  may  infer : 

1.  A  negative  result  indicates  that  some  quantity  in  the  problem 
has  been  applied  in  the  wrong  direction. 

2.  A  possible  problem  analogous  to  the  given  problem  may  be 
formed  by  changing  the  absurd  conditions  to  their  opposites. 

Problems 

435.  Interpret  arithmetically  the  negative  results  obtained  by 
solving  the  following: 

1.  If  A  is  40  years  old  and  B  is  25,  in  how  many  years  will  B 
be  half  as  old  as  A  ? 

2.  Find  the  numbers  whose  sum  is  6  and  difference  10. 

3.  What  fraction  is  equal  to  f  if  1  is  added  to  its  numerator, 
or  to  I  if  1  is  added  to  its  denominator  ? 

4.  A  boy  bought  some  apples  for  24  cents.  Had  he  received 
4  more  for  that  sum,  the  cost  of  each  would  have  been  1  cent  less. 
How  many  did  he  buy  ? 


390  INTERPRETATION  OF  RESULTS 

5.  A  man  worked  7  days,  during  which  he  had  his  son  with 
him  3  days,  and  received  22  shillings.  He  afterwards  worked 
5  days,  during  which  he  had  his  son  with  him  1  day,  aud  received 
18  shillings.     What  were  the  daily  wages  of  each  ? 

ZERO  RESULTS 

436.  When  the  result  obtained  by  solving  a  problem  is  zero,  it 
may  sometimes  indicate  that  the  problem  is  impossible,  and  some- 
times it  may  be  the  proper  answer  to  the  question. 

1.  A  dealer  had  two  kinds  of  tea  worth  75  and  60  cents  per 
pound,  respectively.  How  many  pounds  of  each  must  he  tak-e  to 
make  a  mixture  of  45  pounds  worth  ^  27  ? 

Solution 

Let  X  =  the  number  of  pounds  of  the  better  kind. 

Then,  45  —  x  =  the  number  of  pounds  of  the  poorer  kind ; 

.-.  |x+f(45-a;)=27; 
whence,  x  =  0. 

This  result  means  that  no  such  mixture  can  be  made.  In  fact, 
45  pounds  of  the  poorer  tea  is  worth  $  27. 

2.  A  is  48  years  old,  and  B  is  16  years  old.  After  how  many 
years  will  A  be  3  times  as  old  as  B  ? 

/ 

Solution 

Let  X  =  the  required  number  of  years. 

Then,  48  +  x  =  3(16  +  x). 

Solving,  X  =  0. 

This  result  indicates  that  A  is  noio  3  times  as  old  as  B. 

3.  What  number  is  equal  to  the  square  of  itself  ? 

Solution 
Let  X  =  the  number. 

Then,  x  =  x\ 

x(x-l)  =0; 

.-.  X  =  1  or  0. 

These  results  indicate  that  no  number  except  1  is  equal  to  the 
square  of  itself. 


INTERPRETATION  OF  RESULTS  391 

INDETERMINATE   RESULTS 

437.  1.  A  lady  being  asked  her  age  replied,  "If  from  3  times 
my  age  you  take  4  years  and  divide  the  difference  by  2,  you  will 
have  twice  my  age  less  half  of  my  age  4  years  hence.''  What 
was  her  age  ? 

Solution 
Let  X  =  the  number  of  years. 

Then,  §^Jzi  =  2a:-^±i,  (1) 

'  2  2 

3a;-4=4a;-x-4,  (2) 

(3  -  3)a;  =  0  ;  (3) 

.'.x  =  ^.  (4) 

Since  (2)  may  be  reduced  to  the  identity  3a;  — 4  =  S-r  — 4,  it 
may  be  satisfied  by  any  value  of  x  whatever.  This  relation, 
§  427,  is  expressed  by  (4).     Hence,  the  problem  is  indeterminate. 

Problems 

438.  1.  If  twice  a  certain  number  is  subtracted  from  the 
square  of  the  number,  the  result  will  be  1  less  than  the  square 
of  a  number  1  less.     What  is  the  number? 

2.  A  father  is  30  years  older  than  his  son,  and  the  sum  of 
their  ages  is  30  years  less  than  twice  the  father's  age.  What 
is  the  son's  age? 

3.  The  sum  of  the  first  and  third  of  three  consecutive  integers 
is  equal  to  twice  the  second.     What  are  the  integers  ? 

4.  A  bought  400  sheep  in  two  flocks,  paying  $1.50  per  head 
for  the  first  flock  and  $  2  per  head  for  the  second.  He  lost  30 
of  the  first  flock  and  56  of  the  second,  but  by  selling  the  rest  of 
the  first  flock  at  $  2  per  head  and  the  rest  of  the  second  at  $  2.50 
per  head,  he  neither  lost  nor  gained.  How  many  sheep  were 
there  in  each  flock  originally? 

5.  A  and  B  receive  the  same  monthly  salary.  A  is  employed 
10  months  in  the  j^ear  and  his  annual  expenses  are  $  600.  B  is 
employed  8  months  in  the  year  and  his  annual  expenses  are  $  480. 
If  A  saves  as  much  money  in  4  years  as  B  saves  in  5  years,  what 
is  the  monthly  salary  of  each  ? 


392  INTERPRETATION  OF  RESULTS 


INFINITE  RESULTS 

439.    An  infinite  result  indicates  that  the  problem  is  impossible. 

1.  If  a  man's  yearly  income  is  a  dollars  and  his  yearly  ex- 
penses ar€  a  dollars,  in  how  many  years  will  he  have  saved  b 

dollars  ? 

Solution 

Let     "  X  =  the  required  number  of  years. 

Then,  x  =     ^     =  -,  or  oo. 

a  —  a     0 

That  is,  he  will  never  have  saved  b  dollars  in  this  way. 

2.  A  reservoir  is  fitted  with  three  pipes.  One  pipe  can  fill  the 
reservoir  in  15  hours,  the  second  can  fill  it  in  |  of  that  time,  and 
the  third  pipe  can  empty  it  in  6  hours.  If  the  reservoir  is  full 
and  the  three  pipes  are  opened,  in  what  time  will  it  be  emptied  ? 


, 

Solution 

Let 

X  =  the  required  number  of  hours. 

Then, 

11       1_1 
15      10      6     X 

Solving, 

60 
X  =— ,  or  00. 

That  is,  the  reservoir  will  never  be  emptied  under  these  con- 
ditions. 

3.  What  number  added  to  both  terms  of  the  fraction  ^  will 
make  the  fraction  equal  to  1  ? 

Solution 
Let  X  =  the  number. 

Then,  1±^  =  1. 

2  +  x 

Clearing,  l-{-  x  =  2-\-x. 

Solving,  X  =  -  or  ^^  ; 

that  is,  x  =  +  CO  or  —  00. 

Consequently,  there  is  no  such  number;  but  the  larger  the 
number  in  numerical  value,  the  nearer  will  the  resulting  fraction 
approach  the  value  1. 


INTERPRETATION  OF  RESULTS  393 


THE  PROBLEM  OF  THE  COURIERS 

440.  Two  couriers,  A  and  B,  travel  on  the  same  road  in  the 
direction  from  X  to  Y  at  the  rates  of  m  and  n  miles  an  hour, 
respectively.  At  a  certain  time,  say  12  o'clock.  A  is  at  P,  and  B 
is  at  Q,  a  miles  from  P.     Find  when  and  where  they  are  together. 

X I Y 


Solution 

Suppose  that  time  reckoned  from  12  o'clock  toward  a  later  time  is  positive, 
and  toward  an  earlier  time,  negative  ;  also,  that  distances  measured  from  P 
toward  the  right  are  positive,  and  toward  the  left,  negative. 

Let  X  represent  the  number  of  hours  from  12  o'clock,  and  y  the  number 
of  miles  from  P,  when  A  and  B  are  together.  Then,  they  will  be  together 
y  —  a  miles  from  Q. 

Since  A  travels  mx  miles  and  B  travels  nx  miles  before  they  are  together, 

y  =  mx,  (1) 

and  y  —  a  =  nx.  (2) 

Solving  (1)  and  (2), 

X  = ,  the  required  time.  (3) 

m  —  n  ^  ^  ^ 

y  = ,  the  required  distance.  (4) 


Discussion 

1.  When  a>0  and  my^n. 

When  a  >  0  and  m  >  ?i,  the  numerator  and  denominator  in  (3)  and  also 
in  (4)  are  positive  ;  hence,  x  and  y  are  positive. 

That  is,  A  overtakes  B  some  time  after  12  o'clock,  somewhere  at  the 
right  of  P. 

2.  IVJien  a  >  0  and  m  <  n. 

When  a  >  0  and  m<,n,  both  x  and  y  are  negative. 

That  is,  at  12  o'clock  B  is  ahead  of  A  and  gaining  on  him,  and  they  were 
together  some  time  before  12  o'clock  and  somewhere  at  the  left  of  P. 

3.  When  a  >  0  and  m  =  n. 

When  a  >  0  and  m  =  w,  x  and  y  are  positive  and  infinitely  great. 
That  is,  at  12  o'clock  B  is  ahead  of  A  and  traveling  at  the  same  rate ; 
consequently,  he  will  never  be  overtaken  by  A. 


394"  INDETERMINATE    EQUATIONS 

4.  When  a  =  0  and  m>n  or  m<^n. 

When  a  =  0  and  m  >  n  ov  m<n,  x  =  0  and  y  =  0. 

Lim>n,  x  =  +  0  and  y  =+  0.  That  is,  at  12  o'clock  A  and  B  are  to- 
gether, and  A  is  passing  B, 

If  w  <  w,  x  =  —  0  and  y  =  —  0.  That  is,  at  12  o'clock  A  and  B  are  to- 
gether, and  B  is  passing  A. 

5.  When  a  =  0  and  m  =  n. 

When  a  =  0  and  m  =  n,  x  =  -  and  y  =  — 

That  is,  A  and  B  are  together  at  12  o'clock,  and  since  they  travel  at  the 
same  rate  they  will  be  together  at  all  times. 


:»>*:« 


INDETERMINATE   EQUATIONS 


441.  While  a  problem  that  presents  more  unknown  literal  num- 
bers than  independent  equations  involving  them  is  in  general 
indeterminate  (§  214),  yet  frequently  by  the  introduction  of  a 
condition  or  conditions  not  leading  to  equations,  the  number  of 
values  of  the  unknown  numbers  may  be  limited  and  these  values 
algebraically  determined.  A  common  condition  is  that  the  results 
shall  be  positive  integers. 

1.    Solve  the  equation  5x-\-Sy  =  S5  in  positive  integers. 

Solution 

Since  x  and  y  are  positive  integers,  5  x  must  be  equal  to  6  or  a  multiple 
of  5,  and  3  y  must  be  equal  to  .3  or  a  multiple  of  3.  Since  the  sum  of  these 
multiples  is  35,  if  the  multiples  of  5  are  subtracted  from  35,  one  or  more 
of  the  remainders  will  be  a  multiple  of  3,  if  the  problem  is  possible. 

The  only  multiples  of  5  that  subtracted  from  35  leave  multiples  of  3  are 
5  and  4  times  5. 

.'.  aj  =  1  or  4  ;  whence,  ?/  =  10  or  5. 

Or,  since  x  must  be  a  positive  integer  and  by  transposition  5  a;  =  35  —  3  y, 
the  values  of  x  must  be  1,  2,  3,  4,  5,  or  6,  if  the  equation  is  possible.  Sub- 
stituting these  values  of  x  in  the  given  equation  and  rejecting  all  those  that 
give  negative  or  fractional  values  for  y,  the  positive  integral  values  are  found 
to  be  x  =  1  or  4,  and  2/  =  10  or  5. 


INDETERMINATE   EQUATIONS  395 

2.    Solve  the  equation  5x-\-^y  =  107  in  positive  integers. 

Solution 

5x  +  8y  =  107.  (1) 

Dividing  by  5,  x  +  y  +  ^  =  21  +  |.  (2) 

5 

Collecting  integral  and  fractional  terms, 

x  +  y-21=?.^^.  (3) 

a 

Since  x-\-y  —  2\  is  integral,  ^  =  an  integer.  (4) 

5 

If  ^JZ — y=:w^  an  integer,  then,  y=—^ — ^,  which  is  in  the  fractional 
5  o 

form.     To  avoid  this,  the  coefficient  of  y  in  the  number  placed  equal  to  w 

should  be  made  equal  to  unity.     Since  '—^  is  equal  to  an  integer,  any 

6 
multiple  of  it  is  equal  to  an  integer.     Since  5  is  contained  in  3  times  —By, 
—  2  y  times  with  a  remainder  of  y,  multiplying  (4)  by  3, 


(5) 

(6) 
(7) 
(8) 

Equations  (7)  and  (8)  are  called  the  general  solution  of  the  given  equa- 
tion in  integers. 

To  make  y  and  x  positive  integers,  it  is  evident  from  (7)  that  we  must 
take  10  >  0  ;  and  from  (8)  that  we  must  take  to  <  3. 

Since  w  is  an  integer  greater  than  0  and  less  than  3,  w>  =  1  or  2. 

When  «j  =  1,  X  =  15,  y  =  4  ; 

when  io  =  2,  ^x=    7,  y  =  9. 

3.  Determine  whether  the  equation  10  a;  -|-  15  =  53  may  be 
satisfied  by  integral  values  of  x  and  y. 

Solution.  — Dividing  by  5,  2  x  +  3  y  =  ^. 

If  X  and  y  are  integers,  the  first  member  is  integral. 

Since  the  first  member  is  equal  to  the  fraction  ^^^,  it  cannot  be  an  integer. 
Hence,  x  and  y  cannot  be  integers  at  the  same  time  ;  that  is,  the  equation 
is  not  satisfied  by  integral  values  of  x  and  y. 


^  =  an  integer 

5 

Then,  let 

— i-^  =  w,  an  integer. 

Solving  for  y, 

y  =  bw  —  1. 

Substituting  in 

(3), 

X  =  23  -  8 10. 

S96  INDETERMINATE  EQUATIONS 

Solve  the  following  equations  in  positive  integers : 

4.  5x-\-3y  =  4.9.  8.    12x-\-5y  =  61, 

5.  3  a; +  2  2/ =  5.  9.    6aj  +  7?/    =72. 

6.  2x-\-7y  =  ^S.  10.   5x-\-9y    =75. 

7.  8a;  +  52/  =  80.  11.    6x-{-9y    =100. 

Find  the  least  integral  values  of  x  and  y  in  the  following; 

12.  2x  =  9-\-Sy.  14.    7a;-2y  =  6. 

13.  52/  =  2a;  +  7.  15.    5x-8y  =  l. 

x+y+z=6 

\  in  positive  integers. 


16.   Solve  the  equations    , 

'2x-\-y-z=7 

Solution 
X  +  ?/  +  0  =  6.  (1) 

2x  +  y-z  =  7.  (2) 

Adding,  Sx  +  2y  =  lS.  (3) 

Solving  (3)  for  positive  integers,        x  =  S  and  y  =  2. 
Substituting  in  (1),  z  =  1. 

Solve  the  following  equations  in  positive  integers : 

r3aj  +  22/  =  17, 


f  X  -\-y  +  2j  =  8, 
17.   ^     ^^^  '  19. 

la;  —  ^  +  22  =  6. 


12/4-22  =  14. 


i  3  a;  —  2  =  7. 


2a; +  32/ +  2  =  15, 
3a;  +  2/-2J  =  8. 

21.  Separate  100  into  two  parts  one  of  which  is  a  multiple 
of  11,  and  the  other  a  multiple  of  6. 

22.  In  what  ways  may  a  weight  of  19  pounds  be  weighed  with 
5-pound  and  2-pound  weights  ?  ^ 

23.  A  man  has  $  300  that  he  wishes  to  expend  for  cows  and 
sheep.  If  cows  cost  $  45  apiece  and  sheep  ^  6  apiece,  how  many 
can  he  buy  of  each  ? 

24.  If  9  apples  and  5  oranges  together  cost  52  cents,  what  is 
the  cost  of  one  of  each  ? 


INDETERMINATE  EQUATIONS  397 

25.  A  grocer  sold  two  packages  of  sugar  for  $1.25.  One  pack- 
age contained  a  certain  number  of  pounds  of  7-cent  sugar,  the 
other  a  certain  number  of  pounds  of  5-cent  sugar.  How  many 
pounds  were  there  in  each  package  ? 

26.  A  man  sold  9  animals  —  sheep,  hogs,  and  cows  —  for  $  100. 
If  he  received  $  3  for  a  sheep,  $  6  for  a  hog,  and  $  35  for  a  cow, 
how  many  of  each  did  he  sell  ? 

27.  A  woman  expended  93  cents  for  14  yards  of  cloth,  some  at 

5,  some  at  7,  and  the  rest  at  10  cents  a  yard.     How  many  whole 
yards  of  each  did  she  buy  ? 

28.  Divide  74  into  three  parts  that  shall  give  integral  quotients 
when  divided  by  5,  6,  and  7,  respectively,  the  sum  of  which 
quotients  shall  be  12. 

29.  A  purse  contained  30  coins,  consisting  of  half-dollars, 
quarters,  and  dimes.  How  many  coins  of  each  kind  were  there, 
if  their  aggregate  value  was  $  6.50  ? 

30.  A  man  bought  100  animals  for  $99.  There  were  pigs, 
sheep,  and  ducks.  If  he  paid  $  6  for  a  pig,  $  4  for  a  sheep,  and 
50  cents  for  a  duck,  how  many  of  each  did  he  buy  ? 

31.  What  is  the  least  number  that  will  contain  25  with  a 
remainder  of  1,  and  33  with  a  remainder  of  2  ? 

32.  rind  the  least  number  that  divided  by. 10  and  by  11  will 
leave  remainders  of  3  and  6,  respectively. 

33.  What  is  the  least  number  that  will  contain  2,  3,  4,  5,  and 

6,  each  with  a  remainder  of  1,  and  7  without  a  remainder  ? 

34.  A  man  selling  eggs  to  a  grocer  took  them  out  of  his  basket 
4  at  a  time  and  there  was  1  egg  over.  The  grocer  put  them  into 
a  box  5  at  a  time  and  there  were  3  over.  Both  lost  the  count; 
but  knowing  that  there  were  between  6  and  7  dozen  eggs,  the 
grocer  paid  for  6J  dozen.     How  many  eggs  did  he  lose  ? 

35.  Tour  boys  have  a  pile  of  marbles.  A  throws  away  1  and 
takes  \  of  the  remainder;  B  throws  away  1  and  takes  \  of  the 
remainder;  C  throws  away  1  and  takes  \  of  the  remainder;  D 
throws  away  1,  and  each  boy  takes  \  of  the  remainder.  At  least 
how  many  marbles  must  have  been  in  the  pile,  and  how  many 
does  each  boy  now  have  ? 


THE   BINOMIAL   THEOREM 


442.  The  Binomial  Theorem  derives  a  formula  by  means  of 
which  an}^  power  of  a  binomial  may  be  expanded  into  a  series, 
whether  the  index  of  the  power  is  positive  or  negative,  integral 
or  fractional. 


POSITIVE    INTEGRAL   EXPONENTS 
443.   The  powers  of  (a  +  x),  expanded  in  §  221,  may  be  written 


(a  +  xf  =  a^ 


2ax4-f4^- 

JL  •  ^ 


3.2.1 


3  .  '^ 

(a-hxy  =  a^-\-3a-x  +  —-^ax^  "  1    *>    3 


a^. 


4.3 


(a  +  xy=a^  +  4:  a^x  +  ^-^a^^'  +  - 
(a  +  xY  =  a'  4-  5a^a^  +  f^aV  +  ^ 

X    •  Z  1 


+ 


3.2 


2.3 


aa^  -f 


4.3.2.1   4 

x\ 

1.2.3.4 


4-3^.^^5.4.3.2^^, 


2.3 


1.2.3.4 


4.3.2.1^, 


2.3.4.5 


If  the  law  of  development  revealed  in  the  above  is  assumed  to 
apply  to  the  expansion  of  any  power  of  any  binomial,  as  the  nth. 
power  of  (a  +  x),  the  result  is 

From  formula  (1)  it  is  evident  that  in  any  term, 

1.  The  exponent  of  ic  is  1  less  than  the  number  of  the  term. 
Hence,  the  exponent  of  x  in  the  (r  +  l)th  term  is  r. 

2.  The  exponent  of  a  is  w  minus  the  exponent  of  x. 
Hence,  the  exponent  of  a  in  the  (r  +  l)th  term  is  n  —  r. 


BINOMIAL    THEOREM  399 

3.  The  number  of  factors  in  the  numerator  and  in  the  denomi- 
nator of  the  coefficient  is  1  less  than  the  number  of  the  term. 

Hence,  the  coefficient  of  the  (r  +  l)th  term  has  ?*  factors  in  the 
numerator  and  r  factors  in  the  denominator. 

Therefore,  the  (r  +  l)th,  or  general  term,  is 

n(n  —  1)  (n  —  2)  '"  to  r  factors   „_^  _  ^n^ 

1 -2. 3. .-tor  factors         "        '  ^^ 

Since,  when  there  are  two  factors  in  the  numerator,  the  last  is 
n  —  1,  when  there  are  three  factors,  n  —  2,  when  there  are  four 
factors,  n  —  3,  etc. ;  when  there  are  r  factors,  the  last  is  n  —  (r  —  1), 
or  n  —  r  -f  1.     Hence,  (2)  may  be  written 

n(n-l)(n-2)-"(n-r-{-l)  ,on 

1.2.3...r  "^     '^'  ^'^^ 

Hence,  the  full  form  of  (1)  is 
(a  +  X)'  =  a'  +  na'-'x  +  »(? "i^a-V  +  n(«-l)(«-2)„„-y  ^  ... 

J.   •  Z  1    •  -i   •  O 

n(n-l)(n-2)...(yi-r  +  l)  ,j. 

^  1.2.3-.r  ^^ 

This  is  called  the  Binomial  Formula. 

444.  Since  it  has  already  been  proved,  by  actual  multiplica- 
tion, that  the  binomial  formula  is  true  for  the  second,  third, 
fourth,  and  fifth  powers  of  a'  binomial,  it  remains  to  discover 
whether  it  is  true  for  powers  higher  than  the  fifth. 

If  the  binomial  theorem,  when  assumed  to  be  true  for  the  ntla. 
power,  can  be  j^roved  to  be  true  for  the  (n  +  l)th  power,  it  will 
then  have  been  proved  to  be  true  for  the  sixth  power,  since  it  is 
known  to  be  true  for  the  fifth  power ;  also  for  the  seventh  power, 
being  true  for  the  sixth  power ;  and  in  like  manner  for  each  suc- 
ceeding power. 

It  then  remains  to  prove  that  if  (I)  is  true  for  the  nth.  power, 
it  will  hold  true  for  the  (n  +  l)th  power. 

To  find  the  expansion  of  (a  +  ic)""^^  (I)  may  be  multiplied  by 
a-\-x.  But  since  the  (r  -j-  l)th  term  of  the  product  will  be  the 
algebraic  sum  of  a  times  the  (r  +  l)th  term  of  (a  -f  xy  and  x  times 
the  rth  term  of  (a  +  cc)",  (I)  should  be  prepared  for  multiplication 


400 


BINOMIAL   THEOREM 


by  writing  also  the  rth  term,  obtained  from  the  {r  -f-  l)th  term  by 
substituting  r  for  r  +  1,  or  ?-  —  1  for  r.     Then  (I)  is  written 

(a  +  xy  =  a"  +  na'^-^'x  +  ^^!^T^)  a"" V  +  . . . 

J.  •  ^ 

+  1.2.3...(r-l)  ""        "" 


n(n— l)(n  — 2)-.-(n  — r+2)(n— r+1) 
1.2.3-.-(r-l)r 

Multiplying  both  members  by  a  +  », 


^n-r^.r_| |_^,r 


(a  +  a?) 


n+1  rfM+l 


a"-^'  -f  n 


+  1 


a^icH- 


n  (n  —  1) 
1  .2 


a"-V4-  ••' 


H- 


n(7i- 

-i)(«- 

-2)-(«-r+2)(«- 

r+1) 

+ 

1 

.2.3" 
-1)(« 

.(r-l)r 
-2)...(»- 

r  +  2) 

1.2- 

3...(r-l) 

=  a"+*  -f  (?i  +  1) 


a"a;  +  T- 


(^-1) 


+  n 


a^-^x^+  .. 


=  a"+'  +  (n  + 1)  a"a;  +  ("  +  l)n^,_i^  _!____ 

1    •   ^ 

V     r    J\  1.2.3...(r-l)  ^ 

That  is, 

(a  +  x)«+^  =  a"+i  +  (n  + 1)  a"a;  +  (!?L±_^  a"-iaj2  +  ... 

1  •  iJ 

^  (»»  +  l)n(«-l)-(n-r  +  2)^^„_,^,^ ^  ...  _^ ^„„^    ^j 
1  •  2  •  3  •••  r 

Since  upon  comparison  it  may  be  seen  that  (II)  and  (I)  have 
the  same  form,  n  +  1  in  one  taking  the  place  of  ri  in  the  other, 
(II)  and  (I)  must  express  the  same  law  of  formation. 

Therefore,  if  the  formula  is  true  for  the  nth  power,  it  holds 
true  for  the  (n  +  l)th  power. 


BINOMIAL    THEOREM  401 

Hence,  the  binomial  formula  is  true  for  any  positive  integral 
exponent. 

This  method  of  proof  is  a  proof  by  Mathematical  Induction. 

445.  If  —  ic  is  substituted  for  x  in  (I),  the  terms  that  contain 
the  odd  powers  oi  —  x  will  be  negative,  and  those  that  contain 
the  even  powers  will  be  positive.     Therefore, 

(a  -  xy  =  a"  -  na»-'x  +  ''^''  ~  ^)  a^-'a^ .  (Ill) 

J.  •  j^ 

If  a  =  1,  (I)  becomes 
(1  +  .)-  ^  1  +  nx  +  gfe-7^  u^  +  "^"  7  ^l  ("3-  ^V  +  ■  ■ ..  (IV) 

446.  From  (I)  it  is  seen  that  the  last  factor  in  the  numerator 
of  the  coefficient  is  n  for  the  2d  term,  n  — 1  for  the  3d  term,* 
n  —  2  for  the  4th  term,  n  —  (?i  —  2),  or  2,  for  the  nth  term,  and 
n—(n  —  1),  or  1,  for  the  (n  +  l)th  term ;  and  that  the  coefficient 
of  the  (w  4-  2)th  term,  and  that  of  each  succeeding  term,  contains 
the  factor  n  —  n,  or  0,  and  therefore  reduces  to  0.     Hence, 

Wlien  n  is  a  positive  integer,  the  series  formed  by  eocpanding 
(a  +  a;)"  is  finite  and  has  n  -\-l  terms. 

447.  By  formula  (I),  when  n  is  a  positive  integer, 

(a  +  x)"  =  a»  +  na-'x  +  .'%^„.-V  +  ••  •  +  «(»  -  l)-2  •  1^.. 
^  ^  1-2  1 .2---  (m  —  l)n 

{X  +  ay  =  x'  +  na^-'a  +  fc^x-W  +  ••.  +  fcAlI^a", 

A  comparison  of  the  two  series  shows  that : 
The  coefficients  of  the  latter  half  of  the  expansion  of  (a  +  xy,  when 
n  is  a  positive  integer,  are  the  same  as  those  of  the  first  half,  icritten 

in  the  reverse  order. 

Examples 
1.   Expand  (3a- 26)*. 

Solution.  —  Substituting  3  a  for  a,  2  6  for  x,  and  4  for  n  in  (III), 
(3a-26)*=(3a)4-4(3a)8(2  6)  +  i^(3a)2(2  6)2-i^|^(3a)(2&)8 

^1.2.3.4^      ^ 
=  81  a*  -  216  a%  +  216  a'^h'^  -  96  aft'  +  16  6*. 

.     ACAD.    ALG. 26 


402  BINOMIAL    THEOREM 


2.    Expand  (|  +  6a;  j 


Solution 


7)6 

(1+2  xY  may  be  expanded  by  (IV),  and  the  result  multiplied  by  —  • 


Since  ^1  +  bx^=  [^  <^^  +  ^  ^)]'=  i  ^^  +  ^  ^^' 

(l  +  2xy 

Expand : 
3.    {h-ny. 

4.  (1  +  a-y. 

5.  (2-3x)«. 

6.  (x2-a;)«.  A_aY 

7.  (^  +  c.-y.        ''•  U    yJ'  18.  f^^-^Y- 

8.    (2a+Va;)l  13.    (^a^'  +  ^n^.  ^      ,-      ^    ^-,3 


-  (-!)•■ 

15. 

(^-^J- 

16. 
17. 

«-!            1 

9.    {a  +  a^ay.         14.    (2V2--v/3)«.  '    V^>^2/      3^3 

448.    To  find  any  term. 

Any  term  of  the  expansion  of  a  power  of  a  binomial  may  be 
obtained  by  substitution  in  (2)  or  (3),  §  443. 

In  the  expansion  of  a  power  of  the  difference  of  two  numbers,  as  (a  —  a:)", 
since  the  exponent  of  x  in  the  (r  +  l)th  term  is  r,  the  sign  of  the  general 
term  is  +  if  r  is  even^  and  —  if  r  is  odd. 

Examples 
1.    Find  the  12th  term  of  (a  -  6)". 

Solution 

12th  term^^^-^^-^^-^^-lQ-^-^-^'^'^-^aB(_  ,)n 
1-2. 3. 4. 5-6. 7-8. 9.  10.  11         ^        ^ 

^  _  14  .  13  .  12  ^3^1^  ^  _  3g^  ^3^n. 
1.2.3 

Or 

By  §  447,  since  there  are  15  terms,  the  coefficient  of  the  12th  term,  or  the 
4th  term  from  the  end,  is  equal  to  that  of  the  4th  term  from  the  beginning. 

.-.  12th  term  =  -  ^^  '  ^^  '  ^^  a^h^^  =  -  364  a^b^K 
1.2.3 


BINOMIAL    THEOREM  403 

2.  In  the  expansion  of  (a^  +  2  ic)",  find  the  term  containmg  x^^. 
Solution.  —  Since    (x^  +  2  a:)"  =  fajs/lH-  ^\  l"  =  x^-^  fl  +  -^^   every 

term  of  the  series  expanded  from  [  1  +  -  j     will  be  multiplied  by  x^^. 

^  /2\^        2' 

Hence  the  term  sought  is  that  which  contains  [  -  ]  ,  or  — ;  that  is,  the 

(7  +  l)th,  or  8th  term.  ^^/  ^' 

8th  term  =  x^^  ^^•^^■^■^1  ^V  ^  42240  x^^, 

1 .2.3.4  v«/ 

3.  Find  the  4th  term  of  (a  +  2f.  9  ^  ^    ' 

4.  Find  the  4th  term  of  (x  -  3  ijf.  ^^"HHU  Y  \ 

5.  Find  the  8th  term  of  (a;  -\-yf^.  1  ^0  )c'^^ 

6.  Find  the  5th  term  of  {x-2  y)^.  ^  ^x^  i^'^^  ' 

7.  Find  the  3d  term  of  (a-  -  a'-y.  kp  I'f 

8.  Find  the  20th  term  of  (1  +  xy\         ^2,  .^St)  H  )C 

^  9.   Find  the  16th  term  of  (1  -2x)'^.     —^^f^'^<^  7^  ^ 

10.  Find  the  middle  term  of  (a  +  3  h)\ 

11.  Find  the  6th  term  of  f  a;  +  - )  •  I'^j"- 
r          .                 .                         fx      y^^^ 


/ 12.   Find  the  middle  term  of  ( * — 

/  \y    ^v 

j    13.    Find  the  two  middle  terms  of  /^- -  ^'Y- 

\b      aj 

\  14.    Find  the  coefficient  of  a^  in  the  expansion  of  (a^  +  ay. 

449.  The  formula  given  for  the  expansion  of  (a  +  a:)"  is  true, 
under  certain  conditions,  for  all  commensurable  values  of  v,  whether 
they  are  positive,  negative,  integral,  or  fractional,  and  the  student 
will,  therefore,  be  able  to  expand  such  expressions ;  but  the  proof 
for  negative  and  fractional  exponents  and  the  discussion  of  the 
conditions  under  which  the  expansion  for  these  exponents  gives 
the  true  value  of  (a  +  xy  will  be  deferred.     (See  pages  431-434.) 

In  the  expansion  of  (a  -\-  a;)",  if  n  is  negative  or  fractional, 

none  of  the  binomial  coefficients,  ^  .T  ~I     ?  ^      ~   I      ~^K  etc., 

can  become  0 ;  consequently,  when  such  exponents  are  given,  the 
series  developed  can  have  no  end. 


404  BINOMIAL    THEOREM 

Examples 

1.  Expand  (1  —y)~^  and  find  its  (r  +  l)th  term. 
Solution.  — Substituting  1  for  a,  y  for  ic,  and  —  1  for  n  in  (III), 

(1  -  y)-l  =.  1-1  -(-1)  l-2y  +  ~\^~^^^  1-32/2  -  -K-2)(-3)  ^_4  ^3  _^  ^^^ 
=  1  +  2/  +  2/2  +  ^3    +  .... 

The  (r  +  l)th  term  is  evidently  y. 

Since  (1  —  ?/)-i  = ,  the  above  expansion  of  (1  —  ?/)~i  may  be  verified 

by  division.  ~  ^ 

2.  Expand  {a-\-xy  to  five  terms  and  find  the  10th  term. 

Expand  to  four  terms  : 

3.  (a^h)^.  8.  V4+a;.  13.  (1  +  x)l 

4.  (a  +  6)~l  9.  V(9  -  a;)-''.  14.  (1  +  a)-\ 

5.  (a-6)i  10.  (a^-ic-^)l  15.  (1  _  a)-\ 

6.  A/(a  -  6)3.  11.  (a^-x^y\  16.  (1  -  a^)-2. 

^(a  -  6)3  '""    VV( 

18.  Find  the  (r  +  l)th  term  of  (a  +  x)^. 

19.  Show  that 
(1  _  a;  _  a^)-i  =:l4-a;-}-2a^  +  3a^-t-5x4  +  8.T^4-13a;«  +  21a;^+ 

20.  Find  the  square  root  of  24  to  three  decimal  places. 
Solution.      V24  =(24)^  =(25  -  1)^  =(25)^(1  -  ^^3)^  =  5(1  -  J,)i 

=  5  -  .1  -  .001  -  .00002 =  4.89898  -  =  4.899,  nearly. 

Find  the  values  of  the  following  to  three  decimal  places : 

21.  V5.  23.    V26.  25.    ^9. 

22.  Vrr.  24.    ^/25.  26.    ^30. 


7-    -^7=^==:  12.    (     ,,  ^      _Y-  17.    (l-a;)-3. 

a  —yxj 


LOGARITHMS 


450.    1.    What  power  of  3  is  9  ?     27  ?     81  ?     243  ?     729  ? 

2.  What  power  of  5  is  25 ?    125?    625?    3125?    5?    1?    i? 

3.  Express  100  as  a  power  of  10;  1000  as  a  power  of  10;  10,000 
as  a  power  of  10;  10  as  a  power  of  10;  1  as  a  power  of  10. 

461.  The  exponent  of  the  power  to  which  a  fixed  number  called 
the  Base  must  be  raised  in  order  to  produce  a  given  number  is 
called  the  Logarithm  of  the  given  number. 

When  10  is  the  base,  the  logarithm  of  100  is  2,  for  100  =  102;  the  logarithm 
of  1000  is  3,  for  1000  =  10^  ;  the  logarithm  of  10,000  is  4,  for  10,000  =  10*. 

452.   When  a  is  the  base,  x  the  exponent,  and  m  the   given 
number,  x  is  the  logarithm  of  the  number  m  to  the  base  a. 
It  is  written  log^  m  =  x. 

When  the  base  is  10,  it  is  not  indicated. 

Thus,  the  logarithm  of  100  to  the  base  10  is  2.     It  is  written  log  100  =  2. 

463.  Logarithms  may  be  computed  with  any  arithmetical 
number  except  unity  as  a  base,  but  the  base  of  the  Common  or 
Briggs  System  of  logarithms  is  10. 

Since  10**    =  1,  the  logarithm  of  1  is  0. 
Since  10^    =  10,  the  logarithm  of  10  is  1. 
Since  10^    =  100,  the  logarithm  of  100  is  2. 
Since  10^   =  1000,  the  logarithm  of  1000  is  3. 
Since  10~^  =  -^j  the  logarithm  of  .1  is  —  1. 
Since  lO"''  =  y^,  the  logarithm  of  .01  is  —  2. 

464.  It  is  evident,  then,  that  the  logarithm  of  any  number 
between  1  and  10  is  a  number  greater  than  0  and  less  than  1. 
For  example,  the  logarithm  of  4  is  approximately  0.6021.  Again, 
the  logarithm  of  any  number  between  10  and  100  is  a  number 
greater  than  1  and  less  than  2.  For  example,  the  logarithm  of 
50  is  approximately  1.6990. 

405 


406  LOGARITHMS 

Most  logarithms  are  incommensurable  numbers.  All  the  laws 
established  for  commensurable  exponents  apply  also  to  incom- 
mensurable exponents,  but  the  proofs  have  been  omitted  as  being 
too  difficult  for  the  beginner. 

455.  The  integral  part  of  a  logarithm  is  called  the  Characteris- 
tic ;  the  fractional  or  decimal  part,  the  Mantissa. 

In  log  50  =  1.C990,  the  characteristic  is  1  and  the  mantissa  .6990. 

456.  The  following  examples  will  illustrate  the  characteristic 
and  mantissa,  and  their  significance  : 

log  4580     =  3.6609 ;  that  is,  4580      =  lO^-^^. 
log  458.0    =2.6609;  that  is,  458.0    =  102-««<«. 
log  45.80    =  1.6609 ;  that  is,  45.80    =  lO^-^. 
log  4.580    =0.6609;  that  is,  4.580    =  10«-««». 
log  .4580    =  1.6609;  that  is,  .4580    =  10-i+««». 
log  .0458    =2.6609;  that  is,  .0458    =10-2+-^. 
log  .00458  =  3.6609;  that  is,  .00458  =  10-3+-6««. 
From  the  above  examples  it  is  evident  that : 

457.  Principles.  —  1.  The  characteristic  of  the  logarithm  of  a 
number  greater  than  1  is  positive  arid  1  less  than  the  number  of 
digits  in  its  integral  part. 

2.  The  characteristic  of  the  logarithm  of  a  decimal  is  negative  and 
numerically  1  greater  than  the  number  of  ciphers  immediately  follow- 
ing the  decimal  point. 

458.  To  avoid  writing  a  negative  characteristic  before  a  positive 
mantissa,  it  is  customary  to  add  10  or  some  multiple  of  10  to  the 
negative  characteristic,  and  to  indicate  that  the  number  added  is 
to  be  subtracted  from  the  whole  logarithm. 

Thus,  1.6609  is  written  9.6609  -  10;  2.3010  is  written  8.3010  -  10;  14.9031 
is  written  6.9031  -  20  ;  28.8062  is  written  2.8062  -  30  ;  etc. 

459.  It  is  evident,  also,  from  the  examples,  that  in  the  loga- 
rithms of  numbers  expressed  by  the  same  figures  in  the  same 
order,  the  decimal  parts,  or  mantissas,  are  the  same,  and  that  the 
logarithms  differ  only  in  their  characteristics.  Hence,  tables  of 
logarithms  contain  only  the  mantissas. 


LOGARITHMS  407 

460.  The  table  of  logarithms  on  the  two  following  pages  gives 
the  decimal  parts,  or  mantissas,  correct  to  four  places,  for  the 
common  logarithms  of  all  numbers  from  1  to  1000. 

461.  To  find  the  logarithm  of  a  number. 

Examples 

1.  Find  the  logarithm  of  765. 

Solution.  —  In  the  following  table  the  letter  N  designates  a  vertical 
column  of  numbers  from  10  to  99  inclusive,  and  also  a  horizontal  row  of 
figures  0,  1,  2,  3,  4,  5,  6,  7,  8,  9.  The  first  two  figures  of  765  appear  as  the 
number  76  in  the  vertical  column  marked  N  on  page  409,  and  the  third  figure 
6  in  the  horizontal  row  marked  N. 

In  the  same  horizontal  row  as  76  are  found  the  mantissas  of  the  logarithms 
of  the  numbers  760,  761,  762,  763,  764,  765,  etc.  The  mantissa  of  the  loga- 
rithm of  765  is  found  in  this  row  under  5,  the  third  figure  of  765.  It  is  8837 
and  means  .8837. 

By  Prin.  1,  the  characteristic  of  the  logarithm  of  765  is  2. 

Hence,  the  logarithm  of  765  is  2.8837. 

2.  Find  the  logarithm  of  4. 

,  Solution.  — Although  the  numbers  in  the  table  appear  to  begin  with  100, 
the  table  really  includes  all  numbers  from  1  to  1000,  since  numbers  expressed 
by  less  than  three  figures  may  be  expressed  by  three  figures  by  adding  deci- 
mal ciphers.  Since  4  =  4.00,  and  since,  §  459,  the  mantissa  of  the  logarithm 
of  4.00  is  the  same  as  that  of  400,  which  is  .6021,  the  mantissa  of  the  logarithm 
of  4  is  .6021. 

By  Prin.  1,  the  characteristic  of  the  logarithm  of  4  is  0. 

Therefore,  the  logarithm  of  4  is  0.6021. 

Verify  the  following  from  the  table: 

3.  log  10    =1.0000.  9.  log  .2      =9.3010-10. 

4.  log  100  =  2.0000.  10.  log  542  =2.7340. 

5.  log  110  =  2.0414.  11.  log  345  =2.5378. 

6.  log  2      =0.3010.  12.  log  5.07  =  0.7050. 

7.  log  20    =1.3010.  13.  log  78.5  =  1.8949. 

8.  log  200  =  2:3010.  14.  log  .981  =  9.9917  -  10. 


408 


LOGARITHMS 


Table  of  Common  Logarithms 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

lO 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

II 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

0899 

0934 

0969 

i(X)4 

1038 

1072 

1 106 

13 

"39 

"73 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

37" 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

477^ 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

501 1 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5"9 

5132 

5145 

5159 

5172 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5302 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

39 

59" 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

691 1 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

LOGARITHMS 


409 


Table  of  Common  Logarithms 


N 

0  !  1 

2 

3 

4 

5 

6 

7 

8 

9 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

.8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101  9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154  9159 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206  9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258  9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309  9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

.9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9^§^ 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

98CX) 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

N   0 

1  1  2 

3 

4 

5 

6 

7 

8 

9 

410  LOGARITHMS 

15.  Find  the  logarithm  of  6253. 

Solution.  —  Since  the  table  contains  the  mantissas  not  only  of  the  loga- 
rithms of  numbers  expressed  by  three  figures,  but  also  of  logarithms  expressed 
by  four  figures  when  the  last  figure  is  0,  the  mantissa  of  the  logarithm  of  625 
is  first  found,  since,  §  459,  it  is  also  the  mantissa  of  the  logarithm  of  6250.  It 
is  found  to  be  .7959. 

The  next  greater  mantissa  found  in  the  table  is  .7966,  which  is  the  man- 
tissa of  the  logarithm  of  626  or  of  6260.  Since  the  numbers  6250  and  6260 
differ  by  10,  and  the  mantissas  of  their  logarithms  differ  by  7  ten-thou- 
sandths, it  may  be  assumed  as  sufficiently  accurate  that  each  increase  of  1 
unit,  as  6250  increases  to  6260,  produces  a  corresponding  increase  of  .1  of  7 
ten-thousandths  in  the  mantissa  of  the  logarithm.  Consequently,  3  added 
to  6250  will  add  .3  of  7  ten-thousandths,  or  2  ten-thousandths,  to  the  man- 
tissa of  the  logarithm  of  6250  for  the  mantissa  of  the  logarithm  of  6253. 

Hence,  the  mantissa  of  the  logarithm  of  6253  is  .7959  +  .0002,  or  .7961. 

Since  the  number  is  an  integer  expressed  by  4  digits,  the  characteristic  is 
3  (Prin.  1). 

Therefore,  the  logarithm  of  6253  is  3.7961. 

IJoTE. — The  difference  between  two  successive  mantissas  in  the  table  Is. 
called  the  Tabular  Difference. 

Find  the  logarithm  of 

16.  1054.  20.  21.09.  24.  .09095. 

17.  1272.  21.  3.060.  25.  .10125. 

18.  .0165.  22.  441.1.  26.  54.675. 

19.  1906.  23.  .7854.  27.  .09885. 

462.    To  find  a  number  whose  logarithm  is  given. 

Examples 
1.    Find  the  number  whose  logarithm  is  0.^472. 

Solution.  — The  two  mantissas  adjacent  to  the  given  mantissa  are  .9469 
and  .9474,  corresponding  to  the  numbers  8.85  and  8.86,  since  the  given 
characteristic  is  0.  The  given  mantissa  is  3  ten-thousandths  greater  than 
the  mantissa  of  the  logarithm  of  8.85,  and  the  mantissa  of  the  logarithm  of 
8.86  is  5  ten-thousandths  greater  than  the  mantissa  of  the  logarithm  of  8.85. 

Since  the  numbers  8.85  and  8.86  differ  by  1  one-hundredth,  and  the  man- 
tissas of  their  logarithms  differ  by  5  ten-thousandths,  it  may  be  assumed  as 
sufficiently  accurate  that  each  increase  of  1  ten-thousandth  in  the  mantissa 
is  produced  by  an  increase  of  ^  of  1  one-hundredth  in  the  number.     Conse- 


LOGARITHMS  411 

quently,  an  increase  of  3  ten-thousandths  in  the  mantissa  is  produced  by  an 
increase  of  f  of  1  one-hundredth,  or  .006,  in  the  number. 
Hence,  the  number  whose  logarithm  is  0.9472  is  8.856. 

2.  Find  the  number  whose  logarithm  is  9.4180  —  10. 

Solution-.  —  Given  mantissa,       .4180 

Mantissa  next  less,  .4166  ;  figures  corresponding,  261. 

Difference,  14 

TabuUir  difference,  17)14(.8 

Hence,  the  figures  corresponding  to  the  given  mantissa  are  2618. 
Since  the  characteristic  is  9  —  10,  or  —  1,  the  number  is  a  decimal  with  no 
ciphers  immediately  following  the  decimal  point. 

Hence,  the  number  whose  logarithm  is  9.4180  —  10  is  .2618. 

Find  the  number  corresponding  to 

3.  0.3010.  8.   3.9545.  13.  9.3685-10. 

4.  1.6021.  9.   0.8794.  14.  8.9932-10. 

5.  2.9031.  10.   2.9371.  15.  8.9535-10. 

6.  1.6669.  11.   0.8294.  16.  7.7168-10. 

7.  2.7971.  12.    1.9039.  17.  6.7016-10. 

463.  Multiplication  by  logarithms. 

Since  logarithms  are  the  exponents  of  the  powers  to  which  a 
constant  number  is  to  be  raised,  it  follows  that : 

464.  Principle.  —  The  logarithm  of  the  product  of  two  or  more 
numbers  is  equal  to  the  sum  of  their  logarithms;  that  is, 

To  any  base,  log  (mn)  =  log  m  -\-  log  n. 

The  above  principle  may  be  established  as  follows : 
Let  loga  m  =  X  and  log„  7i  =  y,  a  being  any  base. 

It  is  to  be  proved  that      log.  (mn)  =  x-\-  y. 
§  451,  a'  =  m, 

and  a"  =  n. 

Multiplying,  §  240,  «»+»  =  mn. 

Hence,  §  452,  log,  (mn)  =  x -\-  y 

=  loga  m  +  log.  n. 


412  LOGARITHMS 

Examples 
1.    Multiply  .0381  by  77. 

Solution 
Prin.,  log  (.0381  x  77)  =  log  .0381  +  log  77 

log  .0381  =8.5809- 10 
log77  =  1.8865 


Sum  of  logs  =  10.4674  -  10 
=  0.4674 
0.4674  =  log  2.934. 
.-.  .0381  X  77  =  2.934. 


Note.  — Three  figures  of  a  number  corresponding  to  a  logarithm  may  be 
found  from  this  table  with  absolute  accuracy,  and  in  most  cases  the  fourth 
will  be  correct.  In  finding  the  logarithms  of  numbers  or  the  numbers  corre- 
sponding to  logarithms,  allowance  should  be  made  for  the  figures  after  the 
fourth,  whenever  they  express  .  5  or  more. 

Multiply : 

2.  3.8  by  56.  6.  2.26  by  85.  10.  1.414  by  2.829. 

3.  72  by  39.  7.  7.25  by  240.  11.  42.37  by  .236. 

4.  8.5  by  6.2.  8.  3272  by  75.  12.  2912  by  .7281. 

5.  1.64  by  35.  9.  .892  by  .805.  13.  289  by  .7854. 

465.  Division  by  logarithms. 

Since  the  logarithms  of  two  numbers  to  a  common  base  repre- 
sent exponents  of  the  same  number,  it  follows  that : 

466.  Principle.  —  The  logarithm  of  the  quotient  of  two  numbers 
is  equal  to  the  logarithm  of  the  dividend  minus  the  logarithm  of  the 
divisor;  that  is, 

To  any  base,        log  (m  -r-  w)  =  log  m  —  log  n. 

The  above  principle  may  be  established  as  follows: 

Let  logo  m  =  ic  and  loga  n  =  y^  a  being  any  base. 

It  is  to  be  proved  that      loga(m  -r-  n)=^x  —  y. 

§  451,  a^  =  m  and  ay  =  n. 

Dividing,  §  248,  «*-»  =m  -r-  n. 

Hence,  §  452,  logo(w  ^  n)=x  —  y 

=  loga  m  -  loga  n. 


LOGARITHMS  413 

Examples 

1.  Divide  .00468  by  75. 

Solution 
Prin.,  log  (.00468  -^  75)  =  log  .00468  -  log  75. 

log  .00468  =  7.6702  -  10 
log  75         =  1.8751 
Difference  of  logs  =  5.79r)l  -  10 
5.7951  -  10  =  log  .00006239. 
.-.  .00468  -  75  =  .00006239. 

2.  Divide  12.4  by  16. 

Solution 
Prin.,  log  (12.4  ^  16)  =  log  12.4  -  log  16. 

log  12.4  =  1.0934   =  11.0934  -  10 
log  16     =  1.2041 

Difference  of  logs  =    9.8893  -  10 

9.8893  -  10  =  log  .775.  ^ 

.-.  12.4  -  16  =  .775. 

SuGc.ESTioN.  — The  positive  part  of  the  logarithm  of  the  dividend  may 
be  made  to  exceed  that  of  the  divisor,  if  necessary,  by  adding  10  —  10  or 
20  -  20,  etc. 

Divide : 

3.  3025  by  55.  8.  10  by  3.14.  13.  1  by  40. 

4.  4096  by  32.  9.  .6911  by  .7854.  14.  1  by  75. 

5.  3249  by  57.  10.  2.816  by  22.5.  15.  200  by  .5236. 

6.  .2601  by  .68.  11.  4  by  .00521.  16.  300  by  17.32. 

7.  3950  by  .250.  12.  26  by  .06771.  17.  .220  by  .3183. 

467.    Extended  operations  in  multiplication  and  division. 

Since  dividing  by  a  number  is  equivalent  to  multiplying  by  its 
reciprocal,  for  every  operation  of  division  an  operation  of  multi- 
plication may  be  substituted.  In  extended  operations  in  multi- 
plication and  division  with  the  aid  of  logarithms,  the  latter  method 
of  dividing  is  the  more  convenient. 


414  LOGARITHMS 

468.  The  logarithm  of  the  reciprocal  of  a  number  is  called  the 
Cologarithm  of  the  number. 

The  cologarithm  of  100  is  equal  to  the  logarithm  of  y^^,  which  is  —  2. 
The  cologarithm  of  100  is  —  2  is  abbreviated  to  colog  100  =  —  2. 

469.  Since  the  logarithm  of  1  is  0  and  the  logarithm  of  a 
quotient  is  obtained  by  subtracting  the  logarithm  of  the  divisor 
from  that  of  the  dividend,  it  is  evident  that  the  cologarithm  of 
a  number  is  0  minus  the  logarithm  of  the  number,  or  the  loga- 
rithm of  the  number  with  the  sign  of  the  logarithm  changed ; 
that  is,  if  log„  m  =  x;  then,  colog,,  m  =  —  x. 

Since  subtracting  a  number  is  equivalent  to  adding  it  with  its 
sign  changed,  it  follows  that : 

470.  Principle.  —  Instead  of  subtracting  the  logarithm  of  the 
divisor  from  the  logarithm  of  the  dividend,  the  cologarithm  of  the 
divisor  may  he  added  to  the  logarithm  of  the  dividend;  that  is, 

To  any  base,      log  (m  -^  n)  =  log  m  +  colog  n. 

Examples 

,     -n>-\:i  4-u        1        c  .063  X  58.5  X  799  ,     .        .., 

1.    Find  the  value  of  ^ by  logarithms. 

458  X  15.6  X  .029    ^      ^ 

Solution 

.063  X  58.5  X  799  ^  ^^3  ^  ^g^^  ^  799  x  J-  x  -^  x  J-. 
458  X  15.6  X  .029  458      15.6      .029 

log.  063=    8.7993-10 

log  58.5=    1.7672 

log  799=    2.9025 

colog  458=    7.3391-10 

colog  15.6  =    8.8069  -  10 

colog  .029=    1.5376 

log  of  result  =  31 .  1526  -  30 

=    1.1626. 

.%  result  =  14.21. 


LOGARITHMS  415 


Find  the  value  of 


„     110  X  3.1  X  .653  ^       15  X  .37  x  26.16 


33  x  7.854  X  1.7  11  x  8  x  .18  x  6.67 

_    6000  X  5  X  29         ^   78  X  52  X  1605 

"•         . TTT-Z.*  O. 


.7854  X  25000  x  81.7  338  x  767  x  431 

4       3.516  X  485  X  65  ^       .5  x  .315  x  428 


3.33  X  17  X  18  X  73  .317  x  .973  x  43.7 

471.  Involution  by  logarithms. 

Since  logarithms  are  simply  exponents,  it  follows  that : 

472.  Principle.  —  The  logarithm  of  a  poiver  of  a  number  is 
equal  to  the  logarithm  of  the  number  multiplied  by  the  index  of  the 
power;  that  is, 

To  any  base,  log  m**  =  w  log  m. 

The  above  principle  may  be  established  as  follows : 

Let  logo  m  =  X,  and  let  n  be  any  number,  a  being  any  base. 

It  is  to  be  proved  that  log,  m"  =  nx. 

§  451,  a'  =  m. 

Raising  each  member  to  the  nth  power, 

Ax.  6  and  §  249,  a"*  =  m". 

Hence,  §  462,  logam"  =  nx  =  n  log«TO. 

Examples 
1.    Find  the  value  of  .25^. 

Solution 
IMn. ,  log .  252  =  2  log .  25. 

log  .25  =    9.3979  -  10. 
2  log.  25   =18.7958-20 
=    8.7958  -  10. 
8.7958 -10  =  log  .06249. 
.-.  .25^  =  .06249. 

Note. — By  actual  multiplication  it  is  found  that  .25^  =  .0625,  whereas 
the  result  obtained  by  the  use  of  the  table  is  .06249. 

Also,  by  multiplication,  18-  =  324,  whereas  by  the  use  of  the  table  it  is 
found  to  be  324.1.  Such  inaccuracies  must  be  expected  when  a  four-place 
table  is  used. 


416 

LOGARITHMS 

Find  by  logarithms  the  value  of 

2.    V. 

7.    .781 

12.    4.071 

17. 

(A)'. 

3.    Ill 

8.    8.051 

13.    .5431 

18. 

(^)^- 

4.    471 

9.    8.331 

14.    7*. 

19. 

(tV¥8)^. 

5.    4.91 

10.    Q>.&1\ 

15.    1.02^ 

20. 

(iVA)'- 

6.    5.21 

11.    .7141 

16.    1.7381 

21. 

(2i^)-^. 

473.  Evolution  by  logarithms. 

Since  logarithms  are  simply  exponents,  it  follows  that : 

474.  Principle.  —  Tlie  logarithm  of  a  root  of  a  number  is  equal 
to  the  logarithm  of  the  number  divided  by  the  index  of  the  required 
root;  that  is, 

To  any  base,.  log  vm  = 

The  above  principle  may  be  established  as  follows : 

Let  loga  m  =  X,  and  let  n  be  any  number,  a  being  any  base. 

It  is  to  be  proved  that  loga  V^  =  x  -^  n. 

§  451,  a^  =  m. 

Taking  the  nth  root  of  each  member, 

Ax.  7  and  §  250,  «='-'»  =  Vm. 

Hence,  §  452,  loga  "v^wi  z=lx  -^  n  =  -^ — 

Examples 
1.    Find  the  square  root  of  .1296  by  logarithms. 

Solution 


Prin.,  log  v.  1296  =  ^  log .  1296. 

log.l296  =  9.1126-  10. 

2)19.1126-20 
9.5563  -  10 

9.5563 -10  =  log. 360. 

A  >/l296  =  .36. 


LOGARITHMS 

417 

Find  by  logarithms  the  value  of 

2.    2252. 

9. 

133li 

16. 

V2. 

23. 

^2. 

3.    12.25i 

10. 

1024tV. 

17. 

V3. 

24. 

v:m. 

4.    .2025^ 

11. 

.67241 

18. 

V5. 

25. 

V30T. 

5.    324^. 

12. 

5.929i 

19. 

V6. 

26. 

v:9o. 

6.    .512i 

13. 

.4624^ 

20. 

^2. 

27. 

VI2. 

7.    .118li 

14. 

1.4641^. 

21. 

^4. 

28. 

^.032. 

8.    3.375i 

15. 

.000321 

22. 

^3. 

29. 

V.025. 

Simplify  the 

following : 

176 

oe 

14.5^1:6 

1 

35. 


15  X  3.1416  11 


31  100^  36     ^/•434  X  96^ 

*  48  X  64  X  11  *     >'64  X  1500 

52^^  X  300  .32  X  5000  x  18 

*  12  X  .31225  X  400000'  *    3.14  x  .1222  x  8* 


^,     ^/        400  11  X  2.63  X  4.263 

•    \55  X  3.1416  *  48  x  3.263 


/350C 
\1.06 

40.    2^  X  (i)^  X  ^f  X  V3. 


34.  50  X  I-:-  se     -'^^^«0 


475.    Logarithms  applied  to  the  solution  of  problems  in  compound 
interest  and  annuities. 

1.  What  is  the  amount  of  f  1  for  1  year  at  6  %  ?  By  what 
must  the  amount  for  1  year  be  multiplied  to  find  the  amount  for 
2  years  at  6  %  compound  interest  ? 

2.  By  what  must  the  amount  for  2  years  be  multiplied  to 
obtain  the  amount  for  3  years,  compound  interest?  By  what 
must  the  amount  for  3  years  be  multiplied  to  obtain  the  amount 
for  4  years,  compound  interest  ? 

ACAD.  AI.G.  — 27 


418  LOGARITHMS 

3.  Since  the  amount  of  any  principal  at  6  %  compound  interest 
for  1  year  is  1.06  times  the  principal;  for  two  years,  1.06  x  1.0(5, 
or  1.06^  times  the  principal;  for  3  years,  1.06  x  1.06  x  1.06,  or 
1.06^  times  the  principal,  etc.,  what:  will  be  the  amount  {A)  of  any 
principal  {P)  for  n  years  at  any  rate  per  cent  (r)  ? 

Formula.  A  =  P(l  +  r)\ 

Expressing  the  formula  by  logarithms, 

log  ^  -  log  P  +  7i  log  (1  4-  r).  (1) 

.:\ogP=  log  A  -  n  log  (1  +  r)  ;  (2) 

1  1  /1        .  N  ^Og  ^   —    log   P  .0\ 

also  log  (1  4-  r)  =  — ^^ ^— ,  (3) 

and  ^^^log.l-logP  (4 

log(l+r)  ^^ 

Examples 

1.  What  is  the  amount  of  f  475  for  10  years  at  6%  compound 

interest  ? 

Solution 

A  =  P(l  4-r)« 

log475  1=2.6767 

log  1.0610  =  0.25.30 

log^         =2.9297 

.-.  A  =  $850.60. 

2.  What  will  be  the  amount  of  $  225  loaned  for  5  years  at  8  % 
compound  interest  ? 

3.  Find  the  amount  of  $700  loaned  for  5  years  at  6%  com- 
pound interest. 

4.  Find  the  amount  of  $400  for  10  years  at  3%  compound 
interest. 

5.  Find  the  amount  of  $1200  for  20  years  at  4%  compound 
interest. 

6.  What  principal  will  amount  to  $1000  in  10  years  at  5% 
compound  interest  ? 


LOGARITHMS  419 

7.  What  sum  of  money  invested  at  4%  compound  interest, 
payable  semiannually,  will  amount  to  $  743  in  10  years  ? 

8.  What  principal  loaned  at  4%  compound  interest  will 
amount  to  $  1500  in  10  years  ? 

9.  What  sum  of  money  invested  at  4%  compound  interest 
from  a  child's  birth  until  he  is  21  years  old  will  yield  $  1000  ? 

10.  In  what  time  will  $  800  amount  to  $  1834.50,  if  put  at 
compound  interest  at  5%  ? 

11.  What  is  the  rate  per  cent  when  $  300  loaned  at  compound 
interest  for  6  years  amounts  to  $  402  ? 

12.  A  man  agreed  to  loan  $  1000  at  6%  compound  interest  for 
a  time  long  enough  for  the  principal  to  double  itself.  How  long 
was  the  money  at  interest  ? 

476.  A  sum  of  money  to  be  paid  periodically  for  a  given  number 
of  years,  during  the  life  of  a  person,  or  forever,  is  called  an 
Annuity. 

The  payments  may  be  made  once  a  year,  or  twice,  or  four  times 
a  year,  etc. 

Interest  is  allowed  upon  deferred  payments. 

477.  To  find  the  amount  of  an  annuity  left  unpaid  for  a  given 
number  of  years,  compound  interest  being  allowed. 

1.  If  an  annuity  of  a  dollars  is  not  paid  at  the  end  of  the  first 
year,  how  much  is  then  due  ? 

2.  Upon  what  sum  will  compound  interest  be  computed  for  the 
second  year  ?  What  will  be  the  amount  of  that  sum,  if  the  rate 
is  r  ?  What  will  be  the  whole  sum  due  at  the  end  of  the  second 
year  ?  Ans.  a-\-  a(l  -\-  r). 

3.  Upon  what  sum  will  compound  interest  be  computed  for 
the  third  year?  What  will  be  the  amount  of  that  sum  at  the 
given  rate  ?  '  Ans.  a  (1  +  ?•)  -h  «  (1  +  ^)^- 

What  will  be  the  whole  sum  due  at  the  end  of  the  thi7'd  year  ? 

Ans.  a  +  a  (1  +  r)  +  a  (1  4-  7^y. 


420  LOGARITHMS 

4.  What  will  be  the  whole  sum  due  at  the  end  of  the  fourth 
year?  What  will  be  the  whole  sum  due  at  the  end  of  the  7ith 
year? 

478.  Let  a  represent  the  annuity,  n  the  number  of  years,  r  the 
rate,  and  A  the  whole  amount  due  at  the  end  of  the  nth.  year. 

Then, 

A  =  a  -{-  a{l  -\-  r)  +  a{l  ^  rf  -\-ail-\-  if  -\ \-a(l  +  rf-^ 

=  a  Jl  +  (1  +  r)  +  (1  +  r)2  +  (1  +  rf  ^  .••  +  (1  +  r^-'l 

Since  the  terms  of  A  form  a  geometrical  progression  in  which 
1  +  r  is  the  ratio,  §  370.  the  sum  of  the  series  is 

^="[(1  +  7^-1]. 
?• 

479.  Sometimes  annuities,  drawing  interest,  are  not  payable 
until  after  a  certain  number  of  years.  It  is  often  necessary, 
therefore,  to  find  the  present  value  of  such  annuities. 

480.  A  sum  that  will  amount  to  the  value  of  an  annuity,  if  put 
at  interest  at  the  given  rate  for  the  given  time,  is  called  the 
Present  Value  of  the  annuity.' 

481.  1.  If  P  denotes  the  present  value  of  an  annuity  due  in  n 
years,  allowing  r  %  compound  interest,  to  what  sum  will  P  be 
equal  in  that  time  at  the  given  rate  ?  Aiis.  P(l  +  r)". 

2.  Since  the  amount  of  the  present  value  put  at  interest  for 
the  given  time  at  the  given  rate  is  equal  to  the  amount  of  the 
annuity  for  the  same  time  and  rate,  equate  the  two  sums  and  find 
the  value  of  P. 

P(l-|-r)"  =  ^'[(l +  ,--!]. 
r'     (1  4-  r)" 


(1  +  ry 


LOGARITHMS  421 


Examples 

1.  "What  will  be  the  amount  of  an  annuity  of  $100  remaining 
unpaid  for  10  years  at  6  %  compound  interest  ? 

Solution.  a  =- U\ -\-ry -U, 

r 

By  logarithms,  1.06w  =  1.7904 

.%  1.0610  -  1  =    .7904 

log  100  =  2.0000 

log.7904  =  9.8978 -10 

colog  .06  -  1.2218 

.-.  log^=:  3.1196 

Hence,  -4  =  $  1317,  the  amount  of  the  annuity. 

2.  What  is  the  present  value  of  an  annuity  of  f  100  to  continue 
10  years  at  6%  compound  interest? 

Solution.  P  =  «  .  (^  +  O"  -  ^. 

r        (1  +  r)" 

By  logarithms,  l.Oe^o  =  1.7904 

•*•  1-0610  -  1  3.    .7904 
log  100  =  2.0000 
log.7904  =  9.8978 -10 
colog  .06  =  1.2218 
colog  1.0610  =  9.7470 -10 
.-.  log  P=  2.8666 
Hence,  P=  $735.50,  the  p.  v.  of  the  annuity. 

3.  To  what  sum  will  an  annuity  of  $  25  amount  in  20  years  at 
4%  compound  interest? 

4.  What  is  the  present  value  of  an  annuity  of  $  300  for  5  years 
at  4%  compound  interest? 

5.  What  will  be  the  amount  of  an  annuity  of  $17.76  remain- 
ing unpaid  for  25  years  at  3|%  compound  interest  ? 

6.  AVhat  is  the  present  value  of  an  annuity  of  $  1000  to  con- 
tinue 20  years,  allowing  compound  interest  at  4^%  ? 

7.  What  annuity  will  amount  to  $1000  in  10  years  at  5% 
compound  interest  ? 


UNDETERMINED    COEFFICIENTS 


482.  By  division,  =  1  -\-  x  -\-  x'  -{-  x^  -{-  >> : 

1  —  X 

1.  U  x  =  i,  what  is  the  value  of  the  fraction  ?  What  are  the 
successive  terms  of  the  series,  when  a;  =  i  ?  What  is  the  sum  of 
the  first  2  terms  ?  of  the  first  3  terms  ?  of  the  first  4  terms  ? 
What  value  does  the  sum  of  the  first  n  terms  approach  as  a 
limit  ?  How,  then,  does  the  sum  of  the  series  compare  with  2, 
the  value  of  the  fraction,  whsii  x  =  i  ? 

2.  If  cc  =  1,  what  are  the  successive  terms  of  the  series  ?  How 
large  can  the  sum  of  n  terms  be  ?  How,  then,  does  the  sum  of 
the  series  compare  with  the  value  of  the  fraction  when  x=l? 

3.  How  does  the  sum  of  the  series  compare  with  the  value  of 
the  fraction  when  x  =  2?  when  x=  3? 

4.  If  a?  =  —  1,  what  is  the  value  of  the  fraction  ?  What  is  the 
sum  of  the  first  n  terms  of  the  series  when  n  is  even  ?  the  sum  of 
the  first  71  terms  of  the  series  when  n  is  odd  ?  How,  then,  does 
the  sum  of  the  series  compare  with  the  value  of  the  fraction 
when  X  =  —  1? 

483.  When  the  sum  of  the  first  n  terms  of  an  infinite  series 
approaches  a  finite  number  as  a  limit,  as  n  is  indefinitely  in- 
creased, the  limit  is  called  the  swm*  of  the  series,  and  the  series 
is  called  a  Convergent  Series. 

The  infinite  series  1  +  x  +  x^  -i- x^  +  •",  which  arises  from  the  fraction 

— '■ — ,  is  convergent  if  a;  =  ^  or  any  number  numerically  less  than  1  ;  for  the 

sum  of  the  first  n  terms,  as  n  increases,  approaches  the  value  of  the  fraction 
as  a  limit. 

422 


UNDETERMINED   COEFFICIENTS  423 

484.  When  the  sum  of  the  first  n  terms  of  an  infinite  series  can 
be  made  numerically  greater  than  any  finite  number  by  taking  n 
sufficiently  great,  the  series  is  called  a  Divergent  Series. 

The  infinite  series  1  +  x  +  x^  +  x^  +  •••  is  divergent  if  x  =  1  or  any  number 
numerically  greater  than  1,  as  2,  or  —  2,  or  3,  etc.  j  for  the  sum  of  the  first 
n  terms,  as»w  increases  indefinitely,  becomes  larger  than  any  finite  number. 

When  X  is  numerically  greater  than  1,  the  divergent  series  1  +x+ic^+x''  +  ••■' 
is  not  equal  to  the  fraction  from  which  it  arises. 

"^  485.  When  the  sum  of  the  first  n  terms  of  an  infinite  series 
oscillates  between  certain  fixed  values,  the  series  is  called  an 
Oscillating  Series. 

The  infinite  series  1  +  x  +  x^  +  x'^  +  •••  oscillates  between  the  values  1  and 
0  when  x  =  —  1 ;  for  the  sum  is  1  if  the  number  of  terms  is  odd,  and  0  if  the 
number  of  terms  is  even.  There  is  no  number  of  terms  for  which  the  series 
is  equal  to  the  fraction  from  which  it  arises. 

It  is  evident  tliat  an  oscillating  series  is  neither  convergent  nor  divergent 

It  is  evident  from  the  definitions  given  above,  that  generally 
only  convergent  series  can  be  used  in  demonstrations  and  discus- 
sions involving  infinite  series. 

486.  Coefficients  assumed  in  the  demonstration  of  a  principle 
or  the  solution  of  a  problem,  whose  values,  not  known  at  the 
outset,  are  to  be  determined  by  subsequent  processes,  are  called 
Undetermined  Coefficients. 

487.  Principle  of  Undetermined  Coefficients.  —  If  two 
series  arranged  according  to  the  ascending  powers  of  x  are  equal  for 
every  value  of  x  that  makes  both  series  convergent,  the  coefficients  of 
the  like  powers  of  x  are  equal,  each  to  each. 

Let  A->r  Bx+  Cx-  -\-  Dx^  -\-  —  =■■  A'  +  B'x  +  C"x2  +  D'x"^  +  ...  for  every 
value  of  X  that  makes  both  series  convergent. 

It  is  to  be  proved  that  A  =  A',  B  =  B',   C=  C,  etc. 

Since  when  x  =  0,  the  first  series  is  equal  to  A  and  the  second  is  equal  to 
A' J  each  series  is  convergent  for  x  =  0. 

Since  the  series  are  equal  for  every  value  of  x  that  makes  each  of  them 
convergent,  they  are  equal  for  x  =  0. 

Substituting  0  for  x,  A  =  A'. 

Since  A  and  A'  are  constants  and  equal,  A  =  A',  whatever  the  value  of  x. 

Hence,  Ax.  3,  Bx  +  Cx^  +  Dx^  +  ...  =  B'x  +  C'x'  +  J)'x^  +  -s 


424 


UNDETERMINED    COEFFICIENTS 


Dividing  by  a;,     B  +  Cx  +  Dx^  +  •••  =  JB'+  O'x  +  D'x"^  +  •••. 
Reasoning  as  before,         B  =  B';  C  =  C;  D  =  D'\  etc. 

488.  An  algebraic  expression  is  said  to  be  developed  when  it  is 
transformed  into  a  series  the  sum  of  whose  terms  is  equal  to  the 
given  expression. 

The  series  is  called  the  development  of  the  given  expression. 

When  X  is  numerically  less  than  1,  the  development  of  the  fraction  — - — 
is  the  infinite  converging  series  \  +  x  -\-  x'^  -{■  x^  +  •".  1— x 


DEVELOPMENT  OP  FRACTIONS 
Examples 


489.    1.    Develop  the  fraction 


Assume 


1  -h  a;  +  a^ 
Solution 
'^-^^^    =A-\-Bx+  Cx?-  +  Dx^  +  Ex'^  +  .... 

1  +  X  +  X2 


Clearing  of  fractions  and  collecting  terms, 


1  +  2x 


x2  +  B 
+  C 
+  B 


x^  +  E 
+  D 

+  C 


A  +  B\x+  C 
-Va\     ^-B 
+  A 

Equating  coefl&cients  of  like  powers  of  x,  §  487,  and  observing  that 

1  +  2  X  =  1  +  2  X  +  0  x2  +  0  x3  +  0  x4  +  ..., 

A=\. 

B+  A  =  2 

C+ B  +  A  =  0 

D+  C+  B  =  0 

E+ D+  C  =  0 

.       l+2x 


B  =  l. 

C  =  -2. 
D  =  \. 
E=l. 


1  +  X  -  2  ic2  +  5c3  _h  X* . 

1  +  X  +  x2 

The  fraction  may  be  developed  also  by  division. 

The  exponent  of  x  in  the  first  term  of  the  series  may  always 
be  determined  by  dividing  the  term  in  the  numerator  having  the 
lowest  power  of  x  by  the  term  in  the  denominator  having  the 
lowest  power  of  x.  Beginning  with  this  power  of  x,  a  series  may 
be  assumed  proceeding  according  to  the  ascending  powers  of  x. 


UNDETERMINED   COEFFICIENTS  425 


2.    Develop  the  fraction  2-a;  +  2a^^ 

ar  —  2  ar 

Solution 

o 

Since  the  first,  term  of  the  quotient  is  evidently  — ,  or  2  x-2, 

assume  ^  ~  ^  "^  ^  =  ^ic"^  +  Bx-^  +.C  +  Dx  + Ex^  +  -". 

x2  —  2  a;3 

Clearing  of  fractions, 

2-x  +  2a;'^  =  ^+     B\x  +     C\x^  +    D\x^  +     E\Qd^  + 
-2a\    -2b\      -2C|      -2d\ 

Equating  the  coefficients  of  like  powers  of  oj,  §  487, 

A  =  2. 

B-2A=:-\',     .:  B  =  3. 


C-2B=2 
D-2C=0 
E-2D=0 


.:   0  =  8. 
.-.  D  =  16. 
.-.  E  =  S2. 


.'.  2  -  a;  +  2  x''  ^  2  x-2  +  3  x-i  +  8  +  16  X  +  32  x2  +  .... 
x'^  —  2x^ 

Develop  to  five  terms : 

^3     1±^                      10     l-x-2a^  2~5x 

'   1-x                         '    l-2x-ar'  *    2x-x' 

V  4.    t±l^.                     11           ^-^  18     l±^_+_^. 

'l+a;                          '1+2 a;  — a^  '    x -^ oi^ -\- x*' 

—           ^                            12.     .    ^-^    „.  19.     -1-. 


6.    —^ —  13. -.  20. 


1 

-2a; 

3 

2 

—  X 

1 

1 

—  aa; 

2 

4-3a;2 

1 

-20?^ 

4 

x-Sx" 

7.  ;^^ .14.    -^ -^^'  21. 

1  —  aa;  1  —  .^•  +  2  ar 

8.  ^+3^.  15.    .    ^  +  ^    ,.  22. 


1 

-2a;-ar' 

x-a^ 

l_^2a;-«2 

1-x 

1 

—  x-\-a^ 

1 

1 

-a,--a^ 

2-^x-2x' 

1 

-.^•  +  2ar^ 

a;2  4-.^ 

1 

-2a;  +  a;2 

l-2a; 

(l  —  X 

1 

a  +  a; 

a 

b  —  X 


9      !^Jll__Jl-n.  16  ^       —  23  ^      , 

*     l  +  2a;  •    ic2_|..T3H-.7;^  *    6-aa; 


426  UNDETERMINED   COEFFICIENTS 

DEVELOPMENT  OP    SURDS 
Examples 

490.    1.    Develop  the  expression   Va-f  a;  by  the  use  of  unde- 
termined coefficients. 

.Solution 

Assume  Va  +  x  =  A-\-  Bx-^  Cx^  +  Dx^  +  Ex^  +  •••. 

Squaring,  a  +  x  =  A''-\-'l  ABx  +  B'  j  a;2  +  2  AD  \x^  +  C^\x^  +  .... 

+  '1AC\      +'1Bc\  +  '1Ae\ 

+  2bd\ 

Equating  tlie  coefficients  of  like  powers  of  a:,  §  487, 

A-  =  a  ;    .-.  A  =  Va. 
2^7?=  1;     .-.  5=^. 

B^  +  2AC  =  0;    .-.   C  =  -  ^. 

2AD  +  2BC  =  0;    ..  D  = -^^ 

16  a^ 

C^  +  2AE  +  2BD  =  0;    .'.  E  =  -  ^^ 


128  a* 


.'.  Va  + 


V        2  a      8a2      16a^      128  a*  ; 


The  given  surd  may  also  be  developed  by  the  extraction   of   the  root 
indicated  or  by  the  use  of  the  binomial  formula. 

Develop  the  following  to  five  terms  by  the  use  of  undetermined 
coefficients : 


2.    ^i-x.  8.    Vl  +  a;. 


3.  vi+2ic.  9-  vr+x+^. 


5.  V44-a;.  11.    (l-3x  +  ^x'-xy 

6.  -y/a  —  x.  12.    (1  +  0^)2. 


7.    \(X^-^.  13.    Vl  +  2a;  +  3a^  +  4a,'3+... 


UNDBTrRMINED  COEFFICtENTS  427 


PARTIAL    FRACTIONS 

491.  To  resolve  a  fraction  into  its  partial  fractions  is  to  separate 
it  into  fractions  whose  sum  is  equal  to  the  given  fraction.  Since 
this  is  the  reverse  of  the  process  of  uniting  fractions  with  differ- 
ent denominators  into  a  single  fraction  having  their  lowest 
common  denominator,  it  is  evident  that  the  denominators  of  the 
partial  fractions  must  comprise  all  the  various  factors  of  the 
given  denominator.  The  proper  numerators  may  be  found  by 
the  method  of  undetermined  coefficients. 

Examples 

3 

1.    Resolve into  its  partial  fractions. 

1-bx  +  ^x^  ^ 

Solution 

Assume  that  = 1 is  an  identity. 

1— 5a;^6a;-      1-ox      l-2x 

Clearing  of  fractions,  ^z=A  —  'lAx  +  B  —  Z  Bx. 

That  is,  3  +  0  re  =  ^  +  7i  -  (2  ^  +  3  Z?)  a;. 

.-.  §  487,  ^  +  i?  =  3  and  2  ^1  +  3  5  =  0. 

Solving,  ^  =  9  and  B=-Q. 

3  0  6 


Hence, 


1  —  5a;  +  <3x^      1  —  3a;      1—  2  a; 


2.    Resolve  — — -  into  its  partial  fractions. 

Solution 

Assume  that  — b-Qx —  ^  — A — ^_ —  j^  ^^^  identity. 

I_4x  +  4x2      l-2x      (l-2x)^ 

Reducing,  5-6x  =  ^+  B-2Ax. 

.'.  §  487,  A  +  B  =  5sind  -.2  ^  =  -  6. 

Solving,  A  =  S  and  B  =  2. 

u     r,                              6-6x                3^2 
Hence, = h 


l-4x  +  4a;'-^      1  -2x      (1  -2xy 


428  UNDETERMINED   COEFFICIENTS 

1  —  llx  4-1  x^ 
3.   Kesolve  — — -- — — —  into  its  partial  fractions. 

"  Solution 

By  the  previous  solution  it  is  evident  that  the  fractions  corresponding 
to  the  factor  {\  -  xY  will  be  -^— ,  — — — ,  and         ^ 


\-x    (1  -a:)2'  (1  -xY 

Since  the  factor  (1  +  x  +  a;^)  is  quadratic  and  has  no  rational  simple 
factor,  the  numerator  corresponding  to  the  denominator  1  -f  a:  +  a;-^  may 
have  two  terms  ;  therefore,  assume  that 

l-\\x  +  lx^      _    A      ,        B        ,        C  D  4  Ex        ,j. 


(1  -  x)^(l  +  a;  +  x2)      l-x     (l-a;)2     (l-a;)^     1  +  a;  +  x=^ 
is  an  identity. 

Then,  7  -  11a;  +  7a:2  =  J(l  -  x)2(l  +  x  +  a:2)+  J5(l  -  a:)(l  ^  x -\- x^) 

+  C(l  +  a;  +  x2)  +  (Z)+^x)(l  -  x)^  (2) 

is  an  identity  ;  that  is,  is  true  for  all  values  of  x. 

Since  there  are  five  coefficients,  A^  J5,  C,  Z),  and  E^  to  be  determined, 
and  since  (2)  is  true  for  all  values  of  x,  by  giving  x  in  succession  each  of 
five  different  values,  live  independent  equations  involving  the  undetermined 
coefficients  may  be  formed,  and  from  these  equations  the  coefficients  may 
be  determined. 

Let  X  =  1 ;  then,  (1  —  x),  (1  —  x)^,  and  (1  —  x)^  reduce  to  0,  and  the 
identity  (2)  becomes 

3  =  3C;   .-.  0=1.  (3) 

Let  X  =  0  ;  then,  l  =  A-\-B+C  +  D, 

or,  since  C  =  l,  ^  +  5  +  2>  =  6.  C4) 

Let  X  =  -  1  ;  then,  25  =  4^  +  25  +  l  +  8i>-8J?, 
or,  dividing  by  2,  2  ^  +  ^  +  4  Z>  -  4  ^  =  12.  (5) 

Let  X  =  2  ;  then,  \Z  =  1  A -1  B -\-l  -  D -2  E, 

or  7A-1B-D-2E:=6:  (6) 

Let  X  =  -  2  ;  then,  57  =  27  ^  +  9  i5  +  3  +  27  Z)  -  54  iS:, 
or,  dividing  by  9,  SA  +  B  4- SD  -  6E  =  6.  (7) 

Solving  the  equations,  (4),  (5),  (6),  and  (7),  we  have,  together  with  (3), 
A  =  2,  B  =  0,   e=l,  Z>  =  4,  E  =  2. 

Hence  7-llx  +  7x2       _     2  1  4  +  2x    . 

'        (1  -  X)3(l  +  X  +  X2)         l_x        (1-X)3        1+X  +  X2 


UNDETERMINED   COEFFICIENTS  429 

4.  Eesolve  ~- — -  into  its  partial  fractions. 

1  —  ar 

Suggestion.  —Assume  ^-^^^^  =  _A—  +    B  +  Cx  ^ 
l-x3       1-  X      l  +  x-\-  x^ 

5.  Resolve  - — ^ — - —  into  its  partial  fractions. 

ar  —  1 

Suggestion.  —  When  the  numerator  is  not  of  lower  degree  than  the  de- 
nominator, the  numerator  should  be  divided  by  the  denominator  until  the 
remainder  is  of  lower  degree  than  the  denominator.  The  fractional  part 
of  the  resulting  mixed  expression  may  then  be  resolved  into  partial  fractions, 
and  these  may  be  annexed  to  the  integral  part. 

Resolve  each  of  the  following  into  its  partial  fractions : 

2  r 

6.  . — -.  12. 


7.  o-i-^x  13 


\ 


9.   ± — ±: — ii:i.  15. 


4  10.    —^ ^.^.-r^^  jg^ 


S-Qx  +  x" 

3+4:X 

l-\-Hx  +  Wx' 

Sx 

i-x-ex" 

x-x^ 

l-2.T  +  2«2 

(l-x){l-'2xy 

3.T-2 

or'- 

-iSx 

{X. 

-5)3 

^- 

-5 

X?- 

-1 

2a^  +  9.T  +  ll 

af'  +  4x  +  4 

2-6a;4-6.«2 

1- 

-6a;  +  lla^-6ie3 

49 

i^-Sx)X^-x) 


■^  11     3.T-2  ^^     1-f  2.T  +  3a^  +  2ar^ 

{x-Sf  '  x-x^ 

REVERSION  OF  SERIES 

492.  To  revert  a  convergent  series  of  ascending  powers  of  x  is 
to  express  the  value  of  x  by  means  of  a  series  of  ascending  powers 
of  the  sum  of  the  given  series. 

Let  it  be  required  to  revert  the  series 

y  —  ax  -{-  bx-  -\-  cy?  +  dx'^  4-  •••,  (1) 

in  which  x  has  any  value  that  will  render  the  series  convergent. 
Assume  x  =  Ay  ^- Bf  ^- Cf  ^- Dy'' -^  "'-  (2) 


430 


UNDETERMINED    COEFFICIENTS 


Substituting  this  value  of  x  in  (1),  and  dropping  all  terms 
involving  a  higher  power  of  y  than  the  fourth, 

y'  +  - 


y  = 

a  Ay  4-  aB 

f  +  aC 

f  +  aD 

+  hA' 

+  2bAB 

-\-bB' 

-\-cA^ 

-\-2bAC 
+  ZcA^B 

+  d^^ 

Since  (1)  is  an  identity,  §  487, 
aA  =  l',  .'.  A 

aB-{-bA'  =  0',  .'.  B 

aC-\-2bAB  +  cA'  =  0',         .-.   C 

aD  +  bB-  +  2  6^(7+3  cA^B  +  d^l*  =  0; 


bA^ 
a 

b 

~  of 

-2bAB- 

-cA^ 

2b^- 

-ac 

.-.  D 


aM  _  5  abc  +  5  6^ 


Hence, 

1 
x  =  -y 
a 


62,2 
"12/  4-- 


^'  —  ac  ,     aH  —  5  abc  +  5  6^  4  , 
a'  a! 


(3) 


Examples 

1.    Eevert  the  series  u  =  x 1 — 

^  2       3 


+ 


Solution.  —  Substituting  in  (3)*1  for  a,  —  \  for  &,  \  for  c,  —  ^  for  d,  the 
values  of  the  undetermined  coefficients  in  (2)  are  found  ; 


whence. 


a;-?/  +  i2/2+i?/3+ 2\r  + 


Kevert  the  series  in  the  following  equations : 

2.  2/  =  if +  0.-2  +  0)^  +  a;*  H . 

3.  y  =  x  —  ?>3?-^hx"'  —  lx^^ . 

4.  2/  =  ^'H-2a;2-f  3a;^  +  4a;^H _ 

6.  2/  =  2iK  +  3a;2_,_4^^5^4_|_... 

r^  />*t3  /y»4 

•^  2      6      24 

7.  2/  =  ^—  3a;^  +  5aj^. 

8.  ?/  =  ic-2u;^  +  2ar5-2x*+.... 


UNDETERMINED   COEFFICIENTS  431 

9.    Find  the  approximate  value  of  x  to  four  terms  in  the  series 


x^    ,    x^ 


2     2^12^30      56 

Solution 
Reverting,       x  =  2(^)  -  |(i)2  -  ^^(|)3  _  ^o^(i)4  _  ... 

=  1  ~  i  —  ? jj  —  13^2  —  ••• 
=  .8189+. 

Find  the  approximate  value  of  x  in  the  following : 

10.   i  =  x  +  -  +  — +  —  +••♦. 
2  6      24      60 

5  3  ^  10        7    ^ 


BINOMIAL  THEOREM -FRACTIONAL  AND 
NEGATIVE  EXPONENTS 

493.    It  has  been  shown  (§  445,  formula  IV)  that  when  n  is 
a  positive  integer, 

(H-^)-  =  l+«^  +  ?i^^a-'  +  "("-^)(V^^^+-    (1) 

It  is  yet  to  be  proved  that  this  formula  is  true  when  n  is  a 
positive  fraction,  a  negative  integer,  or  a  negative  fraction. 

I.    When  n  is  a  positive  fraction. 

P 
Let  n  =  -,  in  which  jj  and  q  are  positive  integers. 


Then,  §  246,  (1  +  xy  =  V(l  +  xy 


§444,  =^i^px^....  (2) 


Assume,  §  490,  ■</! +px-{- '■- =  A  + Bx+ Cx''-{- -.-  (3) 

where  x  may  have  any  value  that  will  make  both  series  convergent. 
liaising  both  members  to  the  ^th  power, 

l+px+'-=lA  +  (Bx+Cx'+ •")']'' 

=  A^  +  qA"-' {Bx  +  Cx^  +  •••)  i-  -. 


432  UNDETERMINED    COEFFICIENTS 

Equating  the  coefficients  of  like  powers  of  x  in  two  terms, 
1  =  A'^  and  p  =  qA'-^B ; 

whence,  A  =  l,  and  B  =  -- 

q 

-     Substituting  these  values  in  (2)  and  (3), 

q 

That  is,  the  formula  is  true  for  two  terms,  when  n  is  a  positive 
fraction. 

II.    When  n  is  negative,  and  either  integral  oi^  fractional. 

1 


§244,  (1  +  ^)- 

§  444  and  Case  I, 


(1-hxy 
1 


Dividing  the  numerator  by  the  denominator, 

(1  +  xy  =  1  —  nx -\ . 

That  is,  the  formula  is  true  for  two  terms  when  n  is  negative 
and  either  integral  or  fractional. 

Therefore,  the  formula  is  true  for  two  terms,  whether  the 
exponent  is  positive  or  negative,  integral  or  fractional,  and  the 
coefficient  of  the  second  term  is  n. 

494.    To  find  the  coefficients  of  terms  after  the  second,  assume 
(1  4-  xy=  l-\-7ix-{-Ax'  +  Bx"  4-  Cx^  +  ....  (1) 

Since  only  two  terms  of  the  expansion  of  the  first  member  are 
knoimi,  the  coefficients  A,  B,  etc.,  cannot  be  determined  from  (1) 
in  its  present  form. 

To  find  the  value  of  the  undetermined  coefficients  A,  B,  etc.,  we 
involve  them  in  an  identical  equation,  so  that  the  coefficients  of 
like  powers  of  the  same  variable  may  be  equated  (§  487). 

Since  in  (1)  x  represents  any  number,  put  1  -f-  (a;  +  z)  for  1  -\-x. 


UNDETERMINED    COEFFICIENTS  433 

Then,  (1  +  x  -f-  zy  =  [1  +  {x  +  z)^ 
by  (1),  =l-\-7i{x  +  z)+A(x-\- zf  +  B(x  +  zf  +  - 

-\.  (n  -^2  Ax -^-^  Bx'  +  '■')z -{-  "-.  (2) 

Since  in  (1)  x  represents   any  number,  and   since  1  +  (a;  +  z) 

^(lJ^x)+z  =  (l-^x)(l+-^~\  put  l  +  T-^  for  l+o;. 
Then, 


(l-|-«^  +  ^)"=(l  +  a^)"(l  +  ^J 


Z         ,      A  Z^  .     T>         2^ 


+ 


by  (1),  =(l+.).^l4-»j^^  +  ^^^,  +  B^^_^^^., 

=  (1  +  xY  +  ?i  (1  +  i»)""'2; 

+  ^  (1  +  a;)'-V  +  ^(1  +  xy-^^  +  •-.         (3) 

Therefore,  from  (3)  and  (2),  Ax.  1, 

(1  +  xY  +  n  (1  4-  xy-^  +  ^1  (1  +  »•)"- V  +  i^  (1  +  a;)"-V  +  •  •  • 
^\-\-nx-\-Aa^-\-Bx'^ h  (^^ +  2^a;  +  3  5ar  H )z^ 

when  both  members  are  convergent. 
Equating  the  coefficients  of  «, 

§  487,  •     n{\  +  xy-^  =  71  +  2  Ax  +  3^-2+  .... 

Multiplying  each  member  by  1  +  x, 

n  (1  +  xy  =  71  H-  (2  yl  +  n)  .^•  +  (3  5  +  2  ^)  a.-2  +  . . .. 

(1)  X  n,  n  (1  +  xy  —  n-\-  n^x  +  nAx^  -\ . 

Equating  coefficients,      2  A-{-n  =  n^, 
and  3B-{-2A  =  7iA; 

whence,  ^  =  !?l(!L:z1), 

1.2 

and  ^^7i(n-l)(n-2X 

1-2.3 

ACAD.  ALG.  28 


434  UNDETERMINED   COEFFICIENTS 

In  like  manner  any  number  of  successive  coefficients  may  be 
found.     Substituting  these  values  in  (1), 

The  expansion  of  (1  +  xy  is  not  the  expansion  for  the  most 
general  form  of  a  binomial,  since  the  first  term  is  1 ;  consequently, 
the  equation  must  be  changed  so  as  to  express  the  expansion  of 
(a  -h  xY  for  the  most  general  form.     Putting  -  for  x, 

n 


a)  a         1-2     a'  1.2.3         a' 


(5) 


a"\ 


or  ('^±^Y=  i  (a"  +  na-^x  +  »J^^  a^V  + 


Multiplying  by  a", 

(a  +  xY  =  a^  +  na^-^x  +  ^^^~/^a"-V  +•••.  (6) 

The  binomial  formula  has  thus  been  proved  true  when  n  is  any 
positive  integer,  any  positive  fraction,  any  negative  integer,  or 
any  negative  fraction  ;  that  is,  v^^hen  n  is  any  commensurable  expo- 
nent, provided  the  second  member  of  the  formula  is  convergent 
when  it  is  an  infinite  series. 

495.  It  has  been  shown  (§  449)  that  the  series  developed  from 
{a  +  xY  is  infinite  for  fractional  or  negative  values  of  ii.  It  can 
be  shown  (see  Advanced  Algebra,  §  586)  that  such  a  series  is  con- 
vergent or  divergent  according  as  x  is  numerically  less  or  greater 
than  a.  Hence,  when  x  is  numerically  less  than  a  the  expansion 
of  (a  +  xY  gives  the  true  value  as  near  as  we  please,  but  when  x 
is  numerically  greater  than  a  the  true  value  is  found  by  expanding 
(x  +  a)",  for  the  latter  expression  then  gives  a  convergent  series. 

Thus,  Viol  is  not  found  from  (1  +  100)^  =  1  +  50  -  2500  +  •••,  but  from 
(100  +  1)^  =  10  +  2^  —  5q1^o  +  •••,  which  approaches  VlOl  as  a  limit. 

When  x  =  —  a,  (a  +  a;)~  =  0"  =  0  ;  when  x  =  a,  (a  +  xy  =  (2  a)", 
the  value  of  which  may  be  found  by  separating  2  a  into  a  binomial 
whose  first  term  is  numerically  greater  than  the  second  and  ex- 
panding by  the  binomial  formula.    Thus,  (5  +  5)^  =  10^  =  (9  -f  1)1 

Exercises  for  practice  will  be  found  on  page  404. 


PERMUTATIONS   AND   COMBINATIONS 


496.  All  the  different  orders  in  which  it  is  possible  to  arrange 
a  given  number  of  things,  taking  either  some  or  all  of  them  at 
a  time,  are  called  the  Permutations  of  the  things. 

Thus,  the  permutations  of  the  letters  a  and  h  are  «&,  ha ;  the  permuta- 
tions of  three  letters,  two  at  a  time,  are  a6,  ac,  &a,  6c,  ca,  ch. 

497.  All  the  different  selections  that  it  is  possible  to  make 
from  a  given  number  of  things  taking  either  some  or  all  of  them 
at  a  time,  without  regard  to  the  order  in  which  they  are  placed, 
are  called  the  Combinations  of  the  things. 

Thus,  while  the  pernmtations  of  three  letters,  two  at  a  time,  are  ah  and 
6a,  6c  and  c6,  and  ca  and  ac^  their  comhinations,  two  at  a  time,  are  ah  (or 
6a) ,  6c  (or  c6) ,  and  ac  (or  ca) ;  again,  the  six  permutations  of  three  letters 
among  themselves,  viz.,  a6c,  ac6,  6ca,  6ac,  ca6,  and  c6a,  form  but  one  com- 
bination, a6c  (or  ac6,  or  6ca,  or  6ac,  or  ca6,  or  c6a). 

It  is  evident  that  there  can  be  but  one  combination  of  any  number  of 
things  taken  all  at  a  time. 

498.  Notation.  —  The  symbol  for  the  number  of  permutations 
of  n  different  things,  taken  r  at  a  time,  is  P";  of  n  different 
things,  taken  n  at  a  time,  or  all  together,  P;j. 

The  symbol  for  the  number  of  combinations  of  n  different 
things,  taken  r  at  a  time,  is  O" ;  of  7i  different  things,  taken  w 
at  a  time,  or  all  together,  is  G^- 

499.  The  product  of  the  successive  natural  numbers  from  1 
to  n,  or  from  n  to  1,  inclusive,  is  called  factorial  n,  written  \n. 

[5=1x2x3x4x5,  or  5x4x3x2x1; 

[w=1.2.3-.  (n-2)(/i-l)w,  or  w(w-l)(w-2)(w-3)  .•.3.2.1.^ 

[n  is  sometimes  written  nl 

436 


436  PERMUTATIONS  AND   COMBINATIONS 

500.  To  find  the  number  of  permutations  of  n  different  things  taken 
/•  at  a  time. 

Since  the  permutations  of  the  letters  a,  b,  and  c,  taken  2  at  a 
time,  are  ab  and  ac,  ba  and  be,  ca  and  cb,  formed  by  writing  after 
each  of  the  letters  a,  b,  and  c,  each  of  the  other  letters  in  turn,  the 
number  of  permutations  of  3  different  things  taken  2  at  a  time  is 
3x2. 

The  number  of  permutations  of  n  letters  taken  2  at  a  time  may 
be  found  by  associating  with  each  of  the  n  letters  each  of  the 
n  —  1  other  letters.  Consequently,  the  number  of  permutations 
of  n  different  things  taken  2  at  a  time  is  7i(ii  —  1). 

Since  the  number  of  permutations  of  n  letters  2  at  a  time  is 
n(n  —  1),  if  the  letters  are  taken  3  at  a  time  there  w^ill  be  n  —  2 
letters  each  of  which  may  be  associated  with  each  of  the  n(n  —  1) 
permutations  of  letters  taken  2  at  a  time.  Hence,  the  number  of 
permutations  of  n  different  things  taken  3  at  a  time  is 

7l(7l-l)(w-2). 

Principle  1.  —  The  number  of  permutations  ofn  different  things 
taken  r  at  a  time  is  equal  to  the  continued  product  of  the  natural 
numbers  from  n  to  n  —  {r  —  1)  inclusive.  The  number  of  factors 
is  r.     That  is, 

P"  =  n{n  —  l)(n  —  2)  •  •  •  to  r  factors 

=  n(n  -  l){n  _  2)  .••  (n  -  r  + 1).  (I) 

Multiplying  and  dividing  the  second  member  of  (I)  by 
(n  —  r){n  —  r  —  l)(n  —  ?-  —  2)  •••  2  •  1 ;  that  is,  by  \n  —  r, 

P»=J^.  (II) 

\n  —  r 

It  will  usually  be  more  convenient  to  employ  formula  (I)  in 
solving  numerical  examples;  but  when  simply  algebraic  results 
are  desired,  formula  (II)  will  be  preferable. 

501.  When  r  =  n;  that  is,  when  the  things  are  taken  all  to- 
gether, the  last,  or  ni\\,  factor  in  (I)  is  1.     Consequently, 

Principle  2.  —  The  number  of  permutations  of  n  different  things 
taken  all  at  a  time  is  equal  to  \n.     That  is, 

Pi  =  n{n-l){n-2)  '"  to  n  factors  =  \n.  (HI) 


PERMUTATIONS  AND   COMBINATIONS  487 


Examples 

1.  Three  boys  enter  a  car  in  which  there  are  5  empty  seats. 
In  how  many  ways  may  they  choose  seats  ? 

Solution.  —  Since  the  first  boy  may  choose  any  one  of  5  seats  ;  and  since, 
after  he  has  chosen  one  of  them,  for  each  seat  that  he  may  choose,  the  second 
boy  may  choose  any  one  of  the  4  seats  remaining,  the  greatest  possible  num- 
ber of  ways  in  which  two  of  the  boys  may  be  seated  is  5  x  4. 

Again,  since  after  each  choice  of  seats  made  by  two  of  the  boys  there  will 
be  left  to  the  third  boy  a  choice  of  one  of  the  3  seats  remaining,  the  number 
of  ways  in  which  all  may  choose  seats  is  5  x  4  x  3,  or  60. 

Or,  by  (I),  P«  =  n{n  -  l)(w  -  2)  ...  {n  -  r  +  1), 

P§  =  5  X  4  X  3  =  60. 

2.  How  many  numbers  between  100  and  1000  can  be  expressed 
by  the  figures  1,  3,  5  ? 

Solution. — Since  the  numbers  lie  between  100  and  1000,  each  must  be 
expressed  by  three  figures.  Hence,  the  number  of  numbers  between  100  and 
1000  that  can  be  expressed  by  the  figures  1,  3,  and  5  is  the  same  as  the  num- 
ber of  permutations  of  these  3  figures  taken  3  at  a  time. 

Since,  Prin.  2,  P^  =  [3  =  3  •  2  . 1  =  6, 

there  are  six  such  numbers.    They  are  135,  153,  351,  315,  513,  and  531. 

3.  How  many  permutations  can  be  made  of  the  letters  in  the 
word  Albany,  each  beginning  with  capital  A? 

Solution.  —  Since  A  is  to  be  prefixed  to  each  permutation  of  the  5  other 
letters,  the  required  number  is 

Pj  =  5x4x3x2x1=  120. 

4.  In  how  many  orders  may  4  persons  sit  on  a  bench  ?--,H 

\J      5.    How  many  permutations  may  be  made  of  the  letters  in  the 
word  steam  ?    i->o 

6.  If  10  athletes  run  a  race,  in  how  many  ways  may  the  first 
and  second  prizes  be  awarded  ?    y  .> 

7.  In  how  many  different  orders  may  the  colors  violet,  indigo, 
blue,  green,  yellow,  orange,  and  red  be  arranged  ?      j^-o  ^  o 

8.  There  are  five  routes  to  the  top  of  a  mountain.  In  how 
many  ways  may  a  person  go  up  and  return  by  a  different  way  ?  -2^ 


488  PERMUTATIONS  AND   COMBINATIONS 

502.  To  find  the  number  of  combinations  of  n  different  things  taken 
r  at  a  time. 

Since  two  letters,  as  a  and  6,  have  two  permutations,  ah  and  ha, 
but  form  only  one  combination,  the  number  of  combinations  of  n 
letters  taken  2  at  a  time  is  one  half  the  number  of  permutations 
of  n  letters  taken  2  at  a  time. 

Since  three  letters  taken  3  at  a  time  have  3x2  permutations, 
but  form  only  one  combination,  the  number  of  combinations  of  n 
letters  taken  3  at  a  time  is  obtained  by  dividing  the  number  of 
permutations  of  n  letters  taken  3  at  a  time  by  3x2. 

Since  four  letters  taken  4  at  a  time  have  [4  permutations  but 
form  only  one  combination,  to  obtain  the  number  of  combinations 
of  n  letters  taken  4  at  a  time,  the  number  of  permutations  of  n 
letters  taken  4  at  a  time  must  be  divided  by  [4. 

Hence  it  follows  that : 

Principle  3. —  The  numher  of  comhinations  of  n  different  tilings 
taken  r  at  a  time  is  equal  to  the  numher  of  permutations  of  n  dif- 
ferent things  taken  r  at  a  time,  divided  hy  the  numher  of  permuta- 
tions of  r  different  things  taken  all  together.     That  is, 


or,  by  (II), 


r(r- 

l)(r. 

-2).. 

•  to  7 

'  factors 

n{7i  — 

.!)(« 

-2).. 

>.(n- 

-.  +  1) 

(IV3 

1- 

2.3. 

...  r 

_     ]n 

\7i  —  r 

.^\L 

_       \n 
r  j  n  — 

-  r 

(V) 

503.  Since  for  every  combination  of  r  things  out  of  n  different 
things  there  is  left  a  combination  of  n  —  r  things,  it  follows  that : 

Principle  4.  -^  The  numher  of  comhiriations  of  71  different  things 
is  the  same  when  taken  ?i  —  r  at  a  time  as  when  taken  r  at  a  time. 
That  is, 

c:_.=c:=,--i^-.  (VI) 

\r\n  —  r 


PERMUTATIONS   AND    COMBINATIONS  439 

The  above  principle  may  be  established  as  follows : 
By(V),  ^^r=rzr^j  (1) 

Substituting  n-r  for  r,     C"_,  = 


\r\n- 

-  r 

\n 

\n-r 

■\n-{n- 

-r) 

\n      ■ 

(2) 

\n-r\r  ^  ^ 

Since  the  second  members  of  (1)  and  (2)  are  identical,  C^-r  =  O"- 

The  above  principle  is  useful  in  abridging  numerical  computations. 
Thus,  the  number  of  combinations  of  18  things  taken  16  at  a  time  is  com- 
puted by  Prin.  3  as  follows  : 

^18  ^  18.17-  16-  15.14.13.  12  .  n  .  10.0.8.7  .6  ■5-4.3  ^  ^^g 
^^        1  .  2  .  .3  .  4  .  5  . 6  •  7  .  8  .  9  .  10  .  11  .  12  .  13  .  14  .  15  .  16 

By  Prill.  4,  the  computation  is  abridged  as  follows : 

Examples 

1.  A  man  has  6  friends  and  wishes  to  invite  4  of  them  to 
dinner.     In  how  many  ways  may  he  select  his  guests  ? 

Solution.  —  Since  each  party,  or  combination,  of  4  guests  could  be 
arranged,  or  permuted,  in  [4  ways,  the  number  of  combinations  must  be  — • 
of  the  number  of  permutations  of  G  things  taken  4  at  a  time.  L_ 

Hence,  the  number  of  ways  is 

e«      P^.^p4-6x5x4x3^.^^ 
*         *      1x2x3x4 

2.  A  man  and  his  wife  wish  to  invite  11  of  their  friends,  6  men 
and  5  women,  to  dinner,  but  find  that  they  can  entertain  only  8 
guests.     In  how  many  ways  may  they  invite  4  men  and  4  women  ? 

Solution. — As  in  the  previous  example,  4  men  may  be  selected  from  0 
men  in  15  ways,  and  in  a  similar  manner  4  women  may  be  selected  from  5 
women  in  5  ways. 

Since  when  any  set  of  4  men  has  been  invited,  the  party  of  8  may  be  com- 
pleted by  inviting  any  one  of  5  sets  of  4  women,  the  whole  number  of  differ- 
ent parties  that  it  is  possible  to  invite  is  15  x  5,  or  75.     That  is, 
6        5_6. 5. 4-3      5-4.3.2^^^ 
^*^^^-1.2.3.4''l.2.3.4 


440  PERMUTATIONS  AND    COMBINATIONS 

3.  In  how  many  ways  may  a  baseball  nine  be  selected  from 
12  candidates  ? 

4.  Find  the  value  of  C^«;  of  C'i)  of  C% 

5.  How  many  different  combinations  of  5  cards  can  be  formed 
from  52  cards  ? 

6.  Which  is  the  greater,  C'l  or  C'^?     C\«  or  C^^? 

J  7.  From  11  Republicans  and  10  Democrats  how  many  different 
committees  can  be  selected  composed  of  6  Eepublicans  and  5 
Democrats  ? 

8.  A  man  forgets  the  combination  of  figures  and  letters  by 
which  his  safe  is  opened.  If  they  are  arranged  on  the  circum- 
ferences of  three  wheels,  one  bearing  the  numbers  0  to  9  inclu- 
sive, another  the  letters  A  to  M  inclusive,  and  the  third  the  letters 
N  to  Z  inclusive,  what  is  the  greatest  number  of  trials  he  may 
have  to  make  to  open  the  safe  ? 

9.  From  6  consonants  and  4  vowels  how  many  words  may  be 
formed  each  consisting  of  4  consonants  and  2  vowels,  if  any 
arrangement  of  the  letters  is  considered  a  word  ? 

Solution.  —  The  number  of  combinations  is  Of  x  C^',  and  since  by  per- 
muting the  letters  of  each  combination  [6  words  can  be  formed,  the  number 
of  words  is  C^x  Cjx  [6. 

10.  In  an  omnibus  that  will  seat  8  persons  on  a  side  there  are 
seated  4  persons,  3  on.  one  side  and  1  on  the  other.  In  how  many 
ways  may  12  more  persons  be  seated  ? 

Solution.  —  Since  5  persons  must  take  seats  on  one  side  and  7  persons  on 
the  other,  12  persons  are  to  be  divided  into  two  classes,  6  and  7.  The  number 
of  these  combinations,  formula  (V),  is 

C-/,  or  Of,  =||. 

Since  each  combination  of  5  may  have  [5  permutations  of  the  5  that 
compose  it,  and  each  combination  of  7  may  have  \J_  permutations  each  of 
which  may  be  associated  with  each  of  the  j5  permutations,  the  required 
number  of  ways  is  C^/  x  P^  x  Pj, 

112 
or  l=-x[5x[7=|i2. 

Or,  since  there  are  12  persons  to  be  seated  in  12  seats,  the  number  of  ways 
isP}|  =  |22. 


PERMUTATIONS  AND   COMBINATIONS  441 

"'  11.  Out  of  20  consonants  and  5  vowels  how  many  words  con- 
taining 3  consonants  and  3  vowels  can  be  formed,  if  any  arrange- 
ment of  the  letters  is  considered  a  word  ? 

12.  How  many  different  siuns  can  be  paid  with  a  cent,  a  half- 
dime,  a  dime,  a  quarter,  and  a  dollar  ? 

13.  From  5  boys  and  5  girls  how  many  committees  of  6  can  be 
selected  so  as  to  contain  at  least  2  boys  ? 

14.  A  company  of  a  soldiers  is  joined  by  another  company  of 
h  soldiers.  In  how  many  ways  is  it  possible  to  leave  c  of  them  to 
garrison  the  fort,  dividing  the  rest  into  two  scouting  parties,  one 
of  m,  the  other  of  n  soldiers  ? 

15.  If  (7^  =  2  C?,  find  the  number  of  things. 

\n  \n 

Solution.  —  By  formula  (V),  C"  = — = —  and  Cg  =       ~ 


Since  Cl  =  2C\ 


|5|?i-5  '     |2|n-2 

2lw 


|6|w-5     |2|n-2 

1        ^      1 

|6(w-5      |n-2' 


\n_ 


|n-2  =  |5|n-5. 
:,  or  (n-2)(n-3)(w-4)=5  x4x3x2xl, 


or  (n-2)(n-3)(»-4)=Gx  5x4. 

.-.  n  =  8. 
16.    If  3  C^  =  2  CtS  find  n,  C%  and  C^X\ 

504.    To  find  the  number  of  circular  permutations  of  n  different 
things  taken  /?  at  a  time. 

Suppose  four  letters  a,  6,  c,  d  placed  in  a  fixed  position  around 
a  circle  in  the  order  abed.  Since  the  arrange- 
ment may  be  read  abed,  bcda,  cdab,  or  dabc, 
without  changing  the  direction  in  which  the 
letters  are  read,  it  is  evident  that  each  circular 
permutation  of  4  letters  taken  all  together 
takes  the  place  of  4  permutations  of  the  letters 
all  together. 

^  That  is,  the  number  of  circular  permutations 

of  4  things  taken  all  together  is  one  fourth  of  the  number  of  per- 
mutations of  4  things  taken  all  together. 


442  PERMUTATIONS   AND    COMBINATIONS 

The  whole  number  of  permutations  of  n  things  taken  all 
together  is  [n.  But  if  the  n  things  are  arranged  around  a  circle, 
n  of  these  permutations  may  be  obtained  from  any  circular  per- 
mutation without  disturbing  the  relative  positions  of  the  things. 

Hence, 

Principle  5.  —  The  number  of  circular  jyermutations  of  n  things 

taken  all  together  is  equal  to  —th  of  the  whole  number  of  their  per- 

n 

mutations  taken  all  together.     That  is, 

In 
P^  (circular)  =  L  =  \n  -  1.  (VI 1) 

71 


Examples 

1.  In  how  many  orders  may  6  persons  seat  themselves  around 
a  table  ? 

2.  In  how  many  orders  may  4  gentlemen  and  their  wives  seat 
themselves  around  a  table  ? 

3.  In  how  many  orders  may  4  gentlemen  and  their  wives  seat 
themselves  around  a  table  so  that  each  gentleman  sits  opposite 
his  wife  ? 

4.  In  how  many  orders  may  4  gentlemen  and  their  wives  seat 
themselves  around  a  table  so  that  each  gentleman  sits  opposite 
a  lady  ? 

5.  In  how  many  ways  may  the  colors  violet,  indigo,  blue,  green, 
yellow,  orange,  and  red  be  arranged  on  a  disk,  the  colors  radiating 
from  the  center  ? 

505.  To  find  the  number  of  permutations  of  n  things  taken  n  at 
a  time  when  they  are  not  all  different. 

If,  in  the  permutation  (a,  b,  c,  d,  e,f,  g),  the  letters  b,  d,  and  g 
are  permuted  while  the  other  letters  remain  fixed  in  position,  the 
resulting  number  of  permutations  will  be  the  same  as  the  number 
of  permutations  of  b,  d,  and  g.  If  b,  d,  and  g  are  different  things, 
the  number  of  permutations  resulting  will  be  |3;  but  if  b,  d,  and(/ 
become  alike,  there  will  be  but  1  permutation. 

That  is,  the  number  of  permutations  of  any  number  of  things 


PERMUTATIONS  AND   COMBINATIONS  443 

when  three  of  them  are  alike  is  equal  to  the  number  of  permuta- 
tions of  the  things,  considered  as  all  different,  divided  by  [3  ;  if  4 
of  the  things  are  alike,  by  ^ ;  \i  p  oi  the  things  are  alike,  by  [p. 
Hence,  it  follows  that : 

Principle  G.  —  The  number  of  2)ermutations  of  n  things,  taken 

\n 
all  together,  when  p  of  them  are  alike,  is  —  • 

Since,  if  q  of  the  remaining  71  —  p  different  things  become  alike, 
but  different  from  the  p  like  things,  the  number  of  permutations 
must  be  divided  by  |g;  if  r  others  become  alike,  by  [r;  etc.:  it 
follows  that : 

PiiiNOiPLE  7.  —  The  number  of  permutations  of  n  things,  taken 

all  together,  when  p>  of  them  are  of  one  kind,  q  of  another,  r  of 

[n  ' 
another,  etc.,  is  ■. • 

lp{q[r- 

Examples 

1.  How  many  permutations  may  be  made  with  the  letters  of 
the  word  Mississippi  taken  all  together  ? 

Ill 
Solution.  — The  number  is  , ,  , ,  ,  ^  =  34650. 

[4|4[2 

2.  How  many  permutations  may  be  made  with  the  letters  of 
each  of  the  following  words,  all  at  a  time  in  each  case :  zoology, 
coefficient,  ecclesiastical,  divisibility  ? 

3.  How  many  permutations  may  be  made  with  the  letters  rep- 
resented in  the  product  6i*6V  written  out  in  full  ? 

6.    To  find  the  total  number  of  combinations  of  n  different  things. 

The  number  of  combinations  of  n  different  things  taken  siicces 
sively  1,  2,  3,  •••  n  at  a  time  is  called  the  total  number  of  combina- 
tions of  n  things. 

The  total  number  of  combinations  of  2  things  is 

Ci  4-  C|  =  2  +  1  =  3,  or  2'  -  1. 
The  total  number  of  combinations  of  3  things  is 

Cf+Ci-f  Oi  =  34-3  +  l  =  T,  or  23-1. 


444  PERMUTATIONS  AND   COMBINATIONS 

The  total  number  of  combinations  of  4  things  is 

Ot  +  C|  +  (7^  +  01  =  4  +  6  +  4  +  1  =  15,  or  2*  -  1. 
Principle  8.  —  The  total  number  of  combinations  oj  n  different 
things  is  2"  —  1. 

The  above  principle  may  be  established  as  follows : 
§  445,  when  n  is  a  positive  integer, 

(1  +  ^)n  ^  1  +  ^a;  +  ^^^   ~  ^^X-^  +  -'  +  ^^^  -  ^)C^  -  ^)  -  l^n 

1-2  1  •  2  •  3  •••  n 

Ifa:  =  l,  2n  =  1  -f  n  +  ^'^^  ~  ^>  +  >»•  +  ^'^^  -  ^^n  -  2)  ...  1 

1.2  1.2.3...n 

prin.  3,  =  1  +  o;*  +  o?  +  -  +  ci=i  +  c;^^. 

•     /7«        —  On  _  1 
•  •  ^  total  —  ^  ^' 

Examples 

1.  How  many  different  sums  can  be  paid  with  a  cent,  a  5-cent 
piece,  a  dime,  a  quarter,  a  half-dollar,  and  a  dollar  ? 

Solution.  —Total  C^  =  26  -  1  =  63. 

2.  A  man  has  10  friends.  In  how  many  ways  may  he  invite 
one  or  more  of  them  to  dinner  ? 

3.  How  many  different  quantities  can  be  weighed  by  weights 
of  1  oz.,  1  lb.,  1  lb.,  5  lb.,  and  10  lb.  ? 

4.  How  many  signals  can  be  made  with  7  flags  ? 

5.  By  permuting  the  letters  of  the  word  counter ,  how  many 
permutations  can  be  formed 

(a)  ending  in  er  ? 

(6)  with  n  as  the  middle  letter  ? 

(c)  without  changing  the  position  of  any  vowel  ? 

(d)  beginning  with  a  consonant  ? 

6.  How  many  numbers  can  be  formed  with  the  digits  1,  2,  3, 
4,  3,  2,  1,  so  that  the  odd  digits  always  occupy  the  odd  places  ? 

7.  If  the  number  of  permutations  of  n  different  things  taken 
5  at  a  time  is  equal  to  24  times  the  number  of  permutations  of 
the  same  number  of  things  taken  2  at  a  time,  find  n. 


(1) 


DETERMINANTS 


507.    Solving  the  simultaneous  independent  equations 
'  a^x  +  b^y  =  \, 

we  nave  x  =  -^-^ ^— S    y  =  -^-^ ^-^. 

ttibi  —  aj)i  aib.2  —  a4>i 

Comparing  the  values  of  x  and  y  it  is  observed  that ; 

1.  They  have  the  same  denominator. 

2.  The  numerator  of  the  value  of  x  may  be  formed  from  the 
denominator  by  replacing  the  coefficients  of  x  by  the  correspond- 
ing known  terms  k^  and  k^. 

3.  The  numerator  of  the  value  of  y  may  be  formed  from  the 
denominator  by  replacing  the  coefficients  of  y  by  the  correspond- 
ing known  terms  k^  and  fcg. 

The  common  denominator  0163  —  «2^i  is  called  the  determinant 
of  the  system. 

A  convenient  symbol  for  OrJ).,  —  a^i,  suggested  by  the  arrange- 
ment in  (1)  of  the  coefficients  of  x  and  y  in  two  columns  and  two 
rows,  is 

called  a  determinant  of  the  second  order. 

aib2  —  ajb^  is  called  the  developed  form^  or  the  development,  of 
this  determinant. 

ttj^a  and  —  a.hx  are  called  its  constituents, 
ttj,  a2,  ^1,  62  are  called  its  elements. 

Note.  —  Some  authors  employ  the  terms  eZemew^  and  constituent  with  the 
meanings  here  given  to  constituent  and  element,  respectively. 

445 


446  DETERMINANTS 


508.   To  develop  a  determinant  of  the  second  order. 

The  second  member  may  be  written  &2«i  —  &i«2,  or  —  bia2  +  ai&2,  etc. 


By  definition, 


1.  The  positive  term,  a,&2  or  ftgai,  is  obtained  by  multiplying  the 
element  %  in  the  Jirst  column  and  Jirst  row  by  the  element  62  ^^ 
the  nea:^  column  and  next  row ;  or  by  multiplying  the  element  63  ^'^ 
the  second  column  and  second  row  by  the  element  a^  in  the  precec?- 
m^  column  and  preceding  row. 

The  selection  of  an  element  from  any  column  or  row  before  the 
selection  of  an  element  from  a  preceding  column  or  row  consti- 
tutes an  inversion. 

Then,  the  positive  term  formed  in  the  first  way  presents  no 
inversions,  but  formed  in  the  second  way  presents  two  inversions, 
namely,  the  selection  of  an  element  from  the  second  column 
before  that  of  an  element  from  the  preceding  column,  and  the 
selection  of  an  element  from  the  second  row  before  that  of  an 
element  from  the  preceding  row. 

In  either  case  the  positive  term  of  the  development  presents  an  even 
number  of  inversions. 

2.  The  negative  term,  —  a^^,  or  —  h^ao,  is  obtained  by  multi- 
plying the  element  a.,  in  the  first  column  and  second  row  by  the 
element  b^  in  the  second  column  and  first  row,  and  making  the 
product  negative;  or  by  selecting  the  elements  in  the  reverse 
order  and  making  the  product  negative.  In  the  first  way  there  is 
an  inversion  of  rows,  in  the  second  way,  an  inversion  of  columns. 

In  either  case  the  negative  term  of  the  development  presents  an  odd 
number  of  inversions. 

609.  Any  square  array  of  rt^  elements  arranged  in  n  columns 
and  n  rows  represents  a  determinant  of  the  nth.  order. 

In  harmony  with  the  principles  of  the  preceding  article  a  deter- 
minant of  any  order  is  now  defined  as  a  square  array  of  numbers 
that,  by  common  agreement,  represents  the  algebraic  sum  of  ail 
the  products,  or  constituents,  that  can  be  formed  by  taking  one 
element,  but  not  more  than  one,  from  each  column  and  from  each 


DETERMINANTS 


447 


row,  making  constituents  that  present  an  even  number  of  inver- 
sions positive  and  constituents  that  present  an  odd  number  of 
inversions  negative. 

510.    Development  of  any  determinant. 
«!     hi     Ci 

Let    ttg    62    ^2   ^6  a  determinant  of  the  third  order. 
%    ^3    ^3 

By  the  definition  of  a  determinant,  each  constituent  of  this 
determinant  contains  three  elements  as  factors,  one  and  only  one 
taken  from  each  column  and  from  each  row. 

Hence,  the  constituents  involving  a^  are  a^h^^  and  —  a-^^c^,  the 
latter  being  negative  because  it  presents  one  in\  ^rsion.  There- 
fore, the  sum  of  the  constituents  involving  a^  is 


011^2^3  —  <^i^3C2?  or  <^l 


02    C2 
'^3    C3 


which  may  be  obtained  from  the  given  determinant  by  cancelirig 
or  deleting  the  elements  that  cannot  be  associated  with  Oj, 

Oj    -^i — er 
thus:  (h    ^2    C2 


^3    ^3     ^3 
The  determinant  of  the  next  lower  order 


by  which  a-^  is 


—  0>2 


multiplied  is  called  the  minor  of  the  element  a^.  When  the  minor 
is  given  the  proper  sign,  in  this  case  +  ?  it  is  called  the  co-factor 
of  the  element. 

Similarly,  the  sum  of  the  constituents  involving  ag  is 

derived  by  deleting  the  elements  that  cannot  be  associated  with  a^, 

(jti    h,    Ci 
thus:  as    ^2    ^ 

^i    h    C3 
and  giving  Og  the  sign  —  ,  because  in  each  constituent  a^  is  chosen 
before  an  element  of  the  preceding  row. 


In  this  case,  since  —  a^ 
a^  is  negative. 


1^ 


z=a2X  — 


the  co-factor  of 


448 


DETERMINANTS 


Similarly,  the  sum  of  the  constituents  involving  %  is 

Do       Cg 

Since  each  constituent  of  the  given  determinant  must  involve 
either  Ui,  ag,  or  a^,  we  have  found  all  the  constituents.     Hence, 

Qi     hi 


Ci 

^2 

C2 

&1 

Cl 

&1 

Cl 

C2 

=  «! 

-  Of2 

+  «3 

^ 
h 

C3 

?>3 

C3 

^2 

^2 

C3 

Ci 

<X2 

c, 

«! 

Cl 

«! 

Ci 

^2 

=  -6, 

"     +&2 

-^>3 

as 

C3 

^3 

C3 

a2 

C2 

C3 

The  same  result  is  obtained  by  using  any  column  or  any  row 
of  elements  as  the  first  column  is  used  above. 

For  example,  selecting  the  elements  of  the  second  column, 

a,     61 

^2      ^2 

«3     h      „ 

=  —  a^iC2,  -\-  a^hiC^ + (1-^2'^^ — ^^2^1 — a-J^zC^ + ^2^3^!? 

which  is  the  former  result  differently  arranged. 

The  above  discussion  applies  to  a  determinant  of  any  order. 
Hence, 

*  The  development  of  a  determinant  of  any  order  is  equal  to  the 
algebraic  sum  of  the  products  of  the  elements  of  any  column  or 
row  and  their  respective  cofactors. 

511.  The  minors  corresponding  to  the  elements  a^  a^j  •••, 
bi,  62?  •••)  3-1*6  denoted  by  Ai,  A2,  •••,  Bi,  B2,  •••• 

512.  Number  of  constituents. 

Since  the  co-factors  of  each  of  the  n  elements  in  any  selected 
column  or  row  of  a  determinant  of  the  nth  order  are  determinants 
of  the  (n  —  l)th  order,  a  determinant  of  the  71th  order  has  n  times 
as  many  constituents  as  a  determinant  of  the  (11  —  l)th  order ; 
this,  in  turn,  has  {n  —  1)  times  as  many  constituents  as  a  deter- 
minant of  the  {n  —  2)th  order ;  and  so  on,  until  a  determinant  of 
the  2d  order  is  reached,  which  has  2  constituents.     Hence, 

A  determinant  of  the  nth  order  has  n(n  —  l)(7i  —  2)  •••  2,  or  [n, 
constituents. 


DETERMINANTS 


449 


Examples 

6  9  8 
10  11  12 
14    15    16 

Solution. — Multiplying    the    elements    of    the    first    column   by  their 
co-factors,  and  adding,  the  given  determinant  is  reduced  to 


Develop  the  determinant 


11     12  1       .-19      8j  I    9      8 

15     16    -^1l5     lel+^^lll     12 


=  _  24  -  240  +  280  =  16. 


2.   Develop  the  determinant 


10    11 
14     15 


4 
8 

12 
16 


Solution.  — Proceeding  as  in  Ex.  1,  the  given  determinant  is  reduced  to 


6 

9 

8 

10 

11 

12 

-5 

14 

15 

16 

2 

3      4 

10 

11     12 

+  9 

14 

15     16 

2       3      4 

6       9      8 

14     15     16 


2       3      4 

6      9      8 

10     11     12 


=  16-5.21 ::  ::i-5(-io)|  '^    ^ 


+  9.2 


-3.2 


11 

12 

15 

16 

9 

8 

15 

16 

Since  by  Ex.  1  the  first  determinant  is  equal  to  16,  the  given  determinant 

3      4 
16|-^-^*|ll    12 

i    +"^-«^|l5     16|  +  ^-^*|^    ^ 
9      8 
.11     12 
=  16  +  40  -  600  +  660  +  432  +  648 
=  -  320. 

Develop  the  following  determinants : 


-3(-6)l    J    ,*|-3.10|^    \ 


1512  -  120  -  144  +  360 


*4. 


5. 


4    9    2 

3 

2 

0 

0 

1 

111 

3    5    7 

t% 

6 

4 

1 

1 

0 

4*    2    7 

8    16 

6. 

1 

2 

2 

3 

8. 

0 

3    2     7 

1    7    1 

4 

3 

2 

2 

0 

2    2    1 

3    3    3 

. 

5    15 

3 

4 

2 

5 

2 

0    0    0 

3    2    1 

7. 

0 

3 

1 

2 

9. 

3 

2    2     1 

4    8    3 

0 

1 

2 

1 

2 

12    1 

5    2    1 

2 

0 

2 

7 

1 

3    5     1 

*  For  economy  of  space  the  sign  of  a  negative  element  may  be  written 
above  the'  element. 


450 


DETERMINANTS 


10.  Express  an- —  2  a  —  hmn -\- 2  he -\-mx  —  ncx  as  a  determi- 
nant. 

Solution 

Since  there  are  6,  or  [3,  constituents,  it  is  likely  that  the  required  determi- 
nant is  of  the  third  order,  and  that  the  terms  —  2  a  and  mx  have  a  factor  1 
or  —  1  unexpressed. 


bmn  +  2  &c  -f  mx  —  ncx 

=  a{n  -n  —  2  •\)—h{m  '  n  —  2  •  c)+  x^'i 


a 

m 

c 

n 

1 

m 

c 

m 

c 

-b 

+  x 

b 

n 

1 

2 

n 

2 

n 

n 

1 

X 

2 

n 

2a 


Express  as  determinants : 

11.  25-21.  14.    a^-h^ 

12.  42  +  33.  15.    a  +  x. 

13.  ah  —  cd.  16.    6^  +  1. 

20.  a^i(?/2  -  2/3)  +  ^2{yz  -  2/1)  +  x^iVi  -  2/2). 

21.  ,  a^  —  ahc  —  ahc  +h^  -\-  c^x  —  ahx. 

22.  abc  —  axy  —  acx  +  xyz  +  ahx  —  hh. 


23.  a 


17. 
18. 
19. 


1  —  w  •  c) 


1  -  (ar^  -f  1). 


rp. 


(n^—n). 


3 

a    c 

2     1 

6 

2     1 

6 

2    1     h 

4 

2     c 

-h 

4    2 

c 

+  c 

3     a 

c 

—  a 

Sac 

5 

a    h 

5     a 

h 

5     a 

h 

4     2    c 

REDUCTION  OF  DETERMINANTS 
513.   A  determinant  that  is  equal  to  zero  is  called  a  vanishing 
determinant. 


514. 

1. 

How 

does 

5 

2 

4 
3 

compare  in 

form 

and  value  wit 

5    2 
4    3 

9 

7     4 
2     1 

witii 

7     2 
4     1 

9 

6     9 
8  .4 

with 

6     8    ^ 
9    4 

2     8     1 

2    3    7 

a,    h,     Ci 

Oj     a2    as 

3     5     6 

with 

8    5    3 

9 

a2     62    C2 

with 

hi     b,     63 

7     3 

4 

1 

6 

4 

«3       &3 

C3 

Ci 

C2          Cgi 

2.  How  is  the  value  of  a  determinant  affected  by  changing  the 
rows  into  columns  and  the  columns  into  rows? 

Principle  1.  —  The  value  of  a  determinant  is  not  changed  by 
changing  its  columns  into  rows  and  its  ro2vs  into  columns,  provided 
that  their  order  of  succession  is  not  changed.  • 


DETERMINANTS 


451 


The  above  principle  may  be  established  as  follows : 
Since  the  1st,  2d,  •••,  nth  columns  become  the  1st,  2d,  •••,  nth  rows,  re- 
spectively, and  vice  versa,  the  relative  position  of  the  elements  is  not  changed. 
Therefore,  each  element  of  any  column  or  row  has  the  same  co-factor  as 
before  the  reduction.     Hence,  the  value  of  the  determinant  is  not  changed. 

Corollary.  —  Whatever  is  true  of  the  columns  of  a  determinant 
is  true  of  its  rows,  and  vice  versa. 

515.  1. 


What  is  the  value  of 

0    3    1 

3    2    5 

3    0 

?     of 

0    2     7 

?     of 

0    0    0 

2    0 

0    4    2 

4    16 

2.  What  is  the  value  of  a  determinant  if  all  the  elements  of 
one  column  or  row  are  zeros  ? 

Principle  2.  — A  determinant  that  has  one  or  more  columns  or 
rows  of  zeros  is  equal  to  zero. 

For  since  each  constituent  must  have  for  a  factor  an  element  of  the 
column  or  row  whose  elements  are  zeros,  each  constituent  is  equal  to  zero. 

3     4 


516.  1.  How  does 


5    8 


compare  in  form  and  value  with 


9 
15 


with 


12 
8 


with 


2.  What  is  the  effect  of  multiplying  or  dividing  all  the  elements 
in  a  column  or  row  by  the  same  number  ? 

Principle  3.  —  Multiplying  or  dividing  all  the  elements  in  a 
column  or  row  of  a  determinant  by  the  same  number  multiplies  or 
divides  the  determinant  by  that  number.    (§  85,  §  104,  3.) 

Corollary. — Changing   the   signs  of  all  the  elements 
column  or  row  changes  the  sign  of  the  determinant. 


07ie 


517.   1.    How  are 


and 


formed  from 


How  do  they  compare  with  the  latter  in  value  ? 


2.    Show  that 


1  2 
5  6 
9    4 


3 

7 
10 


3 

7 
10 


3 

7 
10 


Principle  4.  —  The  interchange  of  any  two  columns  or  of  any 
two  roics  of  a  determinant  changes  the  sign  of  the  determiyiant. 


452 


DETERMINANTS 


The  above  principle  may  be  established  as  follows : 

Let  i>  be  a  determinant  of  the  n\\\  order  and  D'  a  determinant  formed  by 
interchanging  any  two  columns  of  D. 

It  is  to  be  proved  that  D'  =—  D. 

By  the  definition  of  a  determinant,  §  509,  the  elements  forming  each  con- 
stituent may  be  selected  from  the  columns  in  any  order  we  please,  taking  one 
but  not  more  than  one  from  each  column  and  row,  provided  each  constituent 
so  formed  is  given  the  proper  sign  showing  the  even  or  odd  number  of  inver- 
sions  of  the  established  order  of  columns  and  rows. 

Then,  let  the  last  two  elements  of  each  constituent  be  chosen  from  the  two 
columns  to  be  interchanged  in  the  order  in  which  these  columns  stand,  giv- 
ing the  result  the  proper  sign.  By  this  method,  when  the  columns  have  been 
interchanged,  each  constituent  will  have  one  more  inversion  than  before, 
namely,  the  inversion  in  the  order  of  the  last  two  columns. 

Hence,  the  sign  of  each  constituent  will  be  changed  by  interchanging  the 
two  columns,  and  by  the  Distributive  Law  this  changes  the  sign  of  D ;  that 
is,  D'  =-D. 

518.  By  changing  places  successively  with  each  of  the  preced- 
ing columns,  any  column  may  be  made  the  leading  column,  pro- 
vided, Prin.  4,  that  when  the  number  of  columns  supplanted  by  the 
advancing  column  is  odd  the  sign  of  the  determinant  is  changed. 

Since  the  same  is  true  of  the  advance  of  any  row  to  the  position 
of  leading  row,  any  element  may  be  brought  to  the  position  of 
leading  element  by  a  proper  number  of  advances  of  its  column  and 
row,  provided  that  the  sign  of  the  determinant  is  changed  when 
the  sum  of  the  number  of  columns  and  the  number  of  rows  preced- 
ing the  column  and  row  in  which  the  element  stands  is  odd. 

Therefore,  since  the  co-factor  of  the  leading  element  is  always 
positive,  the  sign  of  the  co-factor  of  any  element  is  +  when  the  com- 
bined number  of  columns  and  rows  preceding  the  column  and  row  of 
the  element  is  even,  and  —  when  this  number  is  odd. 

Thus,  in  the  determin 
negative  ;  of  5,  positive  ;  of  6,  negative 


1 

2 

3 

4 

5 

6 

7 

8 

9 

the  co-factor  of  4  is  negative  ;  of  2, 
of  7,  positive  ;  etc. 


519.   The  preceding  principle  suggests  a  device  for  developing 
a  determinant  of  the  third  order. 

draw  diagonals,  thus : 


In 


DETERMINANTS 


453 


The  constituents  aifegCs  and  —ajb^c^,  whose  elements  lie  on  the 
diagonals,  are  called  the  principal  diagonal  and  the  secondary  diago- 
nal, respectively.  In  a  determinant  of  the  third  order  the  principal 
diagonal  is  positive  and  the  secondary  diagonal  is  negative. 

To  find  the  other  positive  and  negative  constituents,  by  two 
interchanges  of  columns,  and  again  by  two,  we  have 

h 

The  principal  diagonals  of  these  determinants  are  the  three 
positive  constituents  of  the  given  determinant  and  the  secondary 
diagonals  are  the  three  negative  constituents. 

The  three  equal  determinants  and  their  diagonals  are  written  in 
the  form  ,  +      +     + 


Oi      61 

Ci 

^ 

Ci      tti 

Ci       «! 

ttg   62 

C2 

= 

b2 

C2       02 

= 

C2    ag 

ttg         53 

C3 

bs 

C3     a^ 

C3    as 

in  which  the  principal  diagonals  are  positive  and  the  secondary  di- 
agonals are  negative. 

Caution.  — This  device  does  not  apply  to  determinants  of  a  higher  order 
than  the  third. 


520 


1.  What  is  1 

bhe  value  of 

\  K      K\ 

7    5    5 

7    5 

3    3 

?  of 

8    3    3 
12    2 

?  of 

8    3 
1     2 

10 


2.  Form  other  determinants  each  with  two  columns  or  rows 
alike  or  differing  by  a  constant  multiplier.    What  value  has  each  ? 

Principle  5.  —  If  the  corresponding  elements  in  any  two  columns 
or  rows  of  a  determinant  are  the  same,  or  if  the  elements  in  one  col- 
umn or  row  are  equimultiples  of  the  corresponding  elements  in  the 
other,  the  determinant  is  equal  to  zero. 

The  above  principle  may  be  established  as  follows: 

1.   Let  D  be  a  determinant  having  two  identical  columns  or  rows. 

By  Prin.  4,  if  these  two  columns  or  rows  are  interchanged  the  sign  of  the 
determinant  will  be  changed,  giving  -D.  But  since  the  two  columns  or 
rows  are  identical,  interchanging  them  does  not  change  the  determinant. 

Hence,  D  =  -  D.     But  D  =  -  D  only  when  D  =  0.     Therefore,  Z>  =  0. 


454 


DETERMINANTS 


2.  Let  the  elements  in  one  column  or  row  be  m  times  the  corresponding 
elements  in  another  column  or  row,  and,  Prin.  3,  let  the  determinant  be 
represented  by  mD. 

Then,  as  in  1,  mD  =  —  mD, 

which  is  true  only  when  D  =  0,  for  m  is  not  equal  to  zero. 


521.   To  what  determinant  of  the  second  order  is 


equal  if  ag  =  0  and  ag  =  0  ?  if  6i  =  0  and  Ci  =  0  ?  if  ^^,  =  0  and 
C2  =  0  ?  if  all  the  elements  but  one  in  any  column  or  row  are 
equal  to  zero  ? 

Principle  6.  —  If  all  the  elements  hut  one  in  any  column  or  row 
of  a  determinant  are  equal  to  zero,  the  determinant  is  equal  to  a 
single  determinant  of  the  next  lower  order,  namely,  the  product  of 
the  element  and  its  cof actor. 

For  each  of  the  co-factors  corresponding  to  the  other  elements  in  that 
column  or  row  has  the  coefficient  0,  and  so  becomes  0. 

522.  By  Prin.  6,  any  determinant  may  be  written  as  the  minor 
of  the  element  1  or  —  1  of  a  determinant  of  the  next  higher  order 
equal  to  the  given  determinant,  provided  that  the  other  elements 
in  the  same  column  or  row  as  1  or  —  1  are  zeros. 


Thus, 


a    h 
c     d  ~ 

I    *    * 
0    a    b 

,    or 

ah* 
c    d    * 

,    or 

*    a    b 
Too 

0    c    d 

0    0    1 

*    c    d 

in  which  each  asterisk  stands  for  any  finite  number. 


523.   1.   lfD  = 


a  -{-b 
c  -hd 


showthatZ>=(5a-3c)  +  (5&-3f?). 
—  3  d  as  determinants,  each  having 


2.  Write  5a  —  Sc  and  5 b 
the  same  second  column  as  D. 

3.  Into  what  two  determinants,  then,  may  D  be  resolved  ? 

Principle  7.  —  If  each  element  of  any  column  or  row  of  a  deter- 
minant is  compound,  the  determinant  may  be  written  as  the  algebraic 
sum  of  two  or  more  determinants.     (§  85.) 

524.  It  follows  from  Prin.  7  that  if  two  or  more  determinants 
differ  only  in  the  elements  of  one  column  or  row,  they  may  be 
united  into  a  single  determinant. 


DE  TERM  IN  A  NTS 


455 


Thus, 


5    2    3 

-3    2    3 

2    4    3 

+ 

1     4    3 

= 

1     5    4 

-15    4 

5-3    2    3 

2  +  1    4    3 
1-15    4 


3    4    3 

0    5    4 


5-3    3    10 
6-4    4    20 

8-7     7    30 
is  the  value  of  the  second  determinant  ? 


525.   1.   Separate 


into  two  determinants.    What 


2.    Separate 


5-3 
6-4 

8-7 


into  four  determinants.   What 


is  the  value  of  each  ?    Then,  what  simpler  form  has 


5 

3    10 

6 

4    20 

8 

7    30 

3  10-12 

4  20-16 
7    30-28 

Then,  what 

Principle  8.  —  If  the  elements  of  any  column  of  a  determiiiant 
are  increased  or  diminished  by  the  corresponding  elements  or  by 
equimultiples  of  the  corresponding  elements  of  any  other  column^  the 
value  of  the  determinant  is  not  changed. 

TJie  same  is  true  of  any  two  rows. 

The  above  principle  may  be  established  as  follows : 
ai    bi    •'•    ki 

*2    be  any  determinant, 
a«     On    •"    kn 
and  let  m  be  any  positive  or  negative  number. 

1.    Suppose  that  the  elements  of  the  second  column  are  multiplied  by  m 
and  added  to  the  corresponding  elements  of  the  first  column. 
Then,  by  Prin.  7,  the  resulting  determinant  is  resolved  thus  : 


Let 


ai  +  mbi    bi 
ai  +  mbi    62 


an  +  mbn    bn 


kn 


«i    bi 
aa    bz 


On      bn 


mb\    b\ 
mbi    &2 


mbn    bn 


Prin.  5, 


=  Z>  +  0  =  Z>. 

2.  Let  the  modified  column  be  any  column  after  the  first,  say  the  rth. 
Then,  by  r  interchanges  of  columns  the  modified  column  may  be  made 

the  leading  column,  and  the  determinant  may  be  resolved  as  in  1,  into  Z)  +  0 
or  —  2)  +  0,  according  as  the  number  of  columns  preceding  the  rth  is  even 
or  odd.  In  either  case  by  restoring  the  leading  column  to  its  original  posi- 
tion the  result  obtained  will  be  Z>,  the  given  determinant. 

3.  A  similar  proof  may  he  given  for  modifying  any  row. 


456 


DETERMINANTS 


1.   Evaluate  the  determinant 


Examples 

2    3    4      1 

4  2     12 
112      3 

5  0    3     10 


2 

3 

4     1 

4 

1 

2 

1 

1  2 

2  3 

= 

5 

0 

3  10 

Solution 

a 

10 
8 
3 

10 

« 

1     2     10 
6    3      8 
5    3     10 

= 

i     2     10 

0     9     52 
0     7     40 

=  - 

9    62 

7     40 

10  2 

6    0  3 

1     1  2 

5    0  3 

Explanation.  —  The  aim  is  to  reduce  each  determinant  in  turn  to  a  de- 
terminant of  the  next  lower  order  (Prin.  6)  by  adding  such  multiples  of  the 
elements  of  some  column  or  row  to  the  corresponding  elements  of  one  or 
more  other  columns  or  rows  (Prin.  8)  that  all  of  the  elements  but  one  in  some 
row  or  column  of  the  resulting  determinant  shall  be  zeros.  The  column  or 
row,  multiples  of  whose  elements  are  added  (or  subtracted),  may  be  called 
an  operating  column  or  row  and  is  marked  with  an  asterisk. 

Thus,  selecting  the  third  row  for  an  operator,  we  subtract  3  times  the 
operator  from  the  first  row,  obtaining  T  0  2  10  ;  and  add  2  times  the  operator 
to  the  second  row,  obtaining  6  0  3  8.  The  operator  itself  must  be  brought 
down  unchanged,  in  order  that  the  parts  added  or  subtracted  may  be  vanish- 
ing determinants. 

Since  all  the  elements  except  —1  in  the  second  column  of  the  resulting 
determinant  are  zeros,  this  determinant  (Prin.  6)  is  equal  to  —1  times  its  co- 
factor,  which  is  negative,  because,  §  518,  the  element  —  1  is  preceded  by  ele- 
ments in  an  odd  number  of  columns  and  rows.  Hence,  —  1  times  this  negative 
co-factor  gives  a  positive  determinant  of  the  third  order. 

Continuing  this  process,  the  result  obtained  is  4. 


2.    Show  that 

1  4     7 

2  5     8 

3  6    9 

is  a  vanishing  determinant. 
Solution 

Prin.  8  and  5, 

1  4     7 

2  5     8 

3  6     9 

= 

1 

2 
3 

3    6 

3    6=0. 

3    6 

3.    Evaluate  th 

e  determ 

inant 

1 
2 
3 

2 

1 

2 
4 
5 
4 

2 

3        4        5 
5      7      9 
9     11     13 
5      3      2 
3      5      6 

• 

DETERMINANTS 


457 


Solution 


12  3 

2  4  5 

3  5  9 
2  4  5 
12  3 


4 

5 

7 

9 

1 

13 

= 

3 

2 

6 

6 

1 

0 

0 

0 

0 

2 

0 

1 

1 

1 

3 

1 

0 

1 

2 

= 

2 

0 

1 

5 

8 

1 

0 

0 

1 

1 

1     0    0 

4     7 

1     4     7 

1     1 

0     1     1 

0  111 
10  12 
0  15  8 
0    0     11 


=  -3. 


1    1   1 

1     5    8 

0     1     1 

* 

Evaluate  the  following; 

8  4  6 

4.   2  2  4 

2  3  4 


5. 


6. 


7. 


4  2  12 

2  3  2  6 

3  2  12 

5  6  4  9 

5  2  7  5 

6  3  14 

4  2  13 
6  3  2  5 

4  12  1 

5  2  3  1 

2  112 

3  2  4  6 


10. 


2  4  4  6  1 

2  5  3  12 

3  112  1 
2  1112 
5  2  2  3  1 

2  12  3  3 
12  2  2  4 

3  2  13  2 

2  3  4  0  2 

3  3  2  3  0 


a  1  1  1 
1  tt  1  1 
1  1  a  1 
Ilia 


626.    To  factor  a  determinant. 

Examples 

1.   Factor  Z>  = 


X 

b 

b 

X 

a 

c 

y 

a 

c 

Solution.  —  If  a  =  r,  the  second  and  third  columns  are  identical  and, 
Prin.  5,  the  determinant  vanishes.  Hence,  by  the  Factor  Theorem,  §  136, 
a  —  c  is  a  factor  of  D. 


458 


DETERMINANTS 


Again,  if  x  =  —  y,  the  second  and  third  rows  are  identical  and  D  =  0. 
Hence,  x  +  ?/  is  a  factor  of  D. 

Since  every  constituent  of  D  is  of  the  third  degree  and  (a  —  c)  (x  +  y)  is 
of  the  second  degree,  D  must  have  another  factor  of  tlie  first  degree.  Substi- 
tuting 0  for  &,  D  is  equal  to  x  times  the  co-factor  ,  which  is  equal  to  0 ; 

that  is,  Z>  =  0.  Hence,  the  other  factor  of  Z>  is  6  —  0,  or  &  ;  or  it  may  be  —h. 
It  remains  to  find  whether  D  =  b  {a  —  c)  {x  +  y)  or  —h  {a  —  c)  {x  -{■  y). 
The  secondary  diagonal  of  D  is  +  aby  and  this  is  the  only  constituent  of 
D  involving  «,  &,  and  y.  Since  the  sign  of  ahy  is  +  in  &  (a  —  c)  (x  +  y) 
but  —  in  —  6  (rt  —  c)  (x  +  y),  Z)  =  &  (a  —  c)  (x  +  y). 

Factor  the  following  determinants  by  inspection: 


2. 


X 

1 

h 

y 

1 

a 

X 

1 

a 

a" 

a 

1 

2 

2 

5 

b' 

b 

1 

4. 

2 

X 

5 

c' 

c 

1 

X 

3 

5 

527    Solution  of  simultaneous  simple  equations. 

It  has  been  shown  that  in  a  system  of  two  simultaneous  simple 
equations  of  the  form  ax  -{-by  =  c,  either  unknown  number  is  equal 
to  a  fraction  ivhose  denoininator  is  the  determinant  of  the  system  and 
whose  numerator  is  the  determinant  of  the  system  with  the  known 
terms  substituted  for  the  cor^^esponding  coefficients  of  that  unknown 
number. 

By  trial,  the  principle  is  found  to  hold  for  the  solution  of  three 
simultaneous  simple  equations. 

'  «ia^  +  biy  +  ciz  =  ^•l, 

Thus,  given  «2X  +  b-zy  +  c^z  =  kz, 

asx  +  bay  +  Csz  =  ks. 

Solving  by  the  ordinary  process  of  elimination,  then  rearranging  and 
grouping  terms, 

^_  ki  (b^cz  -  63C2)  -  k<2,  jbiCz  -  bzC\)  +  kz  (bic^  -  52Ci) 
di  (62C3  -  63C2)  -  a2  (61C3  -  63C1)  +  as  ibiC2  —  62C1) 


^1 

hi 

Cl 

k2 

&2 

C-2 

ks 

b. 

Cs 

«1 

61 

Cl 

«2 

62 

C2 

as 

&8 

Cs 

Similarly  for  the  values  of  y  and  3. 


DE  TERMINA  NTS 


459 


The  principle  will  now  be  proved  to  be  general 


Let 


f  aix  +  hxy  +  Ciz  +  ...  =  A:i, 
a2X  +  622/  +  coz  +  •••  =  A;2, 

anX  +  bnV  +  CnZ  +  •••  =  ^n 


(1) 


be  a  system  of  n  simple  equations.  Let  D  represent  the  determinant  of  the 
system,  Dx  the  determinant  of  the  system  with  the  known  terms  ki,  k2,  ••• 
substituted  for  the  corresponding  coefficients  of  x,  and  ^1,  Ao,  •••  the  co-fac- 
tors of  ai,  a2,  •••• 


Then, 


an    hn    c„ 


and  Dx 


kn    br, 


C2) 


Multiplying  the  first  equation  of  the  system  by  Ai,  the  second  by  A2,  etc., 
and  adding  the  resulting  equations, 

(ai^i  +  02^2  +  •••  +  anAn)x 


=  kiAi  +  A:2^2  + 1-  knAr 


(3) 


-f  (61^1  +  62^2  +  •••  +  bnAn)  y 

+ 

Since  the  coefficient  of  x  in  (3)  is  the  sum  of  the  products  of  the  elements 
in  one  column  of  D  and  their  co-factors,  the  coefficient  of  x  is  equal  to  D, 
and  the  second  member  of  (3)  is  equal  to  Dx-  The  coefficient  of  y  in  (8) 
differs  from  that  of  x  only  in  having  the  elements  of  the  second  column  of  D 
repeated  in  the  first  column,  61,  62,  •••  replacing  ai,  a^  •••,  thus : 

61  61     Ci 

62  bi    C2 


bn      bn      Cn      • 

By  Prin.  6,  this  determinant  is  equal  to  zero,  and  in  like  manner  the  coeffi 
cients  of  the  other  unknown  numbers  vanish.     Hence,  (3)  becomes 

D 


Dx  =  Dx',  wlience,  x 


Similarly,  Dy  =  Dy\  whence,  y  = 

So  for  each  unknown  number. 

Examples 
Solve  the  following  by  determinants : 

f2x-\-5y  =  9, 


D. 


(4) 


3. 


2. 


3a: -f- 2// =  12, 
4a;-h3?/  =  17. 


4.     <! 


2x  +  7y=:S0, 
.T  -I-  4  2/  =  17. 

(5x-\-    y  =  12, 


[2x-{-Sy  =  10. 


460 


DETERMINANTS 


.    Ux-3y  =  S, 


Sx-2y=-2, 
5. 


[2x-Sy= 

.  ( ax-\-by  =  Cf 
[mx-{-ny  =  a. 

(ax  —  by=rj 
[cx  +  dy  =  s. 

(2x  +  5y-{-2z  =  27, 

9.  ■!  Sx-\-6y-^3z  =  A6, 

[3x-\-7y-\-5z  =  l7. 


{  (a  -f  b)x  —  (a  —  b)y  =  4  ab, 
{{ct-b)x+{a  +  b)y  =  2d'-2b\ 

{^3x-^2y  +  3z  =  ll, 
12.  \2x  +  y^2z  =  10, 
[^x^by  +  z  =  2^. 


13. 


r2r<;  +  3?/-42  =  18, 

x  +  y-^z  =  12, 


5x  —  y  —  z  =  12. 

(x-\-2y  +  z  =  0, 


(9x-{-2y  +  z  =  25, 
.      5x-\-y-\-z  =  14, 
[7x-\-Syh2z  =  25. 


16. 


17. 


14.  i2x-\-y-{-z  =  2a  —  b, 
[x  —  y  —  2z  =  3b. 

(x  +  y  =  2a,   ■ 

15.  \y-\-z  =  3a  —  b, 
[z-\-x  =  3a. 

u  —  x-\-2y  —  3z=—5, 
3u  —  x-\-y  —  2z  =  2y 
2u-{-x-]-y  —  z  =  9, 
-5u  +  2x-7  y-\-z= -12. 

2u-}-3v-4:X-\-y  =  0, 
\u  —  v  +  x-y=-2, 
7  u  +  2v-3x-\-y  =  6, 


5u-{-Sv-10x  +  3y  =  3. 

528.  An  equation  in  which  every  term  is  of  the  first  degree  in 
some  unknown  number  is  called  a  Homogeneous  Linear  Equation. 

az  =  by,  or  ax  —  by  =  0,  isa  horaogeueous  linear  equation. 

529.  By  §  527  the  denominator  of  the  value  of  each  unknown 
number  in 

a^x-\-  6,2/4-0,2+  ...=0, 

a<fc  -\-  boy  -\-  c^z  +  •••  =  0,  (1) 


[anX-^b^  +  c^z  +  "-  =  0 


DETERMINANTS 


461 


is  the  determinant  of  the  system,  and  the  numerator  is  the  same 
determinant  with  0  substituted  for  each  coefficient  of  the  unknown 
number.  Therefore,  the  numerator  in  each  case  will  have  one 
column  composed  entirely  of  zeros,  and,  Prin.  2,  will  be  equal  to  0. 


X  = 


0 


y 


,  etc. 


Hence,  each  unknown  number  is  equal  to  zero,  except  when 
D  =  0,  in  which 'case  each  is  indeterminate  and  the  system  is 
indeterminate. 

The  case  in  which  Z>  =  0  is  the  case  in  which  the  equations  in 
(1)  are  not  independent.  For  if  it  is  possible  to  form  any  equa- 
tion in  (1)  by  combining  multiples  of  two  or  more  of  the  other 
equations  by  addition  or  subtraction,  it  is  possible  to  make  two 
rows  of  D  identical  by  the  same  process. 

Hence,  if  n  liomocfeneous  linear  equations  involving  n  unknown 
numbers  are  independenty  the  unknown  numbers  are  sejjarately  equal 
to  zero. 

530.  A  system  of  w  —  1  independent  linear  equations  involving 
n  unknown  numbers  is  indeterminate  (§  214,  proof) ;  but  if  the 
equations  are  homogeneous,  the  ratio  of  any  two  unknown  num- 
bers may  be  found. 


Thus,  let  aix  +  hiy  +  ciz  =  0, 

and  aix  +  b^y  +  co^s  =  0 

be  given,  to  find  the  ratios  of  xioy,  x  to  s,  and  y  to  z. 

From  (1), 


z         z 


Ci. 


From  (2), 


Solving, 


also,  from  (5), 


z         z 


C2. 


-ci  6i 

a\ 

-ci 

-C2  6, 

and  ^  = 

z 

a% 

-C2 

a\    hx 

a\ 

bx 

02  62 

at 

&2 

(1) 

(2) 

(3) 
(4) 

(5) 


Ci     61 

C2 62^ 

a\     -  ci 
a2     —  C2 


I  62    C2 

Ci      «i 

Co    ai 


oc'-jc*' ^  a^  -   yr^  ( VAJtlVytaLtfeukc^  Y^    yr-S\M  iUoJ^    W^^'JL 


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